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Search: a161345 -id:a161345
Displaying 1-10 of 38 results found. page 1 2 3 4
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A161346 a(n) = A161345(n)/3. +20
6
3, 4, 5, 6, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Union of {4, 6, 9} and all the odd primes. - Amiram Eldar, Apr 17 2024
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
MATHEMATICA
Select[Range[271], Function[{n, s}, Max[TakeWhile[Divisors[n], # <= s &]] == 3] @@ {#, Sqrt@ #} &[3 #] &] (* Michael De Vlieger, Feb 14 2020 *)
CROSSREFS
Cf. A018253, A160811, A160812, A161205, A161344, A161345.
KEYWORD
easy,nonn
AUTHOR
Omar E. Pol, Jun 20 2009
EXTENSIONS
Terms beyond a(10) from R. J. Mathar, Jun 24 2009
STATUS
approved
A008578 Prime numbers at the beginning of the 20th century (today 1 is no longer regarded as a prime). +10
548
1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
1 together with the primes; also called the noncomposite numbers.
Also largest sequence of nonnegative integers with the property that the product of 2 or more elements with different indices is never a square. - Ulrich Schimke (ulrschimke(AT)aol.com), Dec 12 2001 [Comment corrected by Farideh Firoozbakht, Aug 03 2014]
Numbers k whose largest divisor <= sqrt(k) equals 1. (See also A161344, A161345, A161424.) - Omar E. Pol, Jul 05 2009
Numbers k such that d(k) <= 2. - Juri-Stepan Gerasimov, Oct 17 2009
Also first column of array in A163280. Also first row of array in A163990. - Omar E. Pol, Oct 24 2009
Possible values of A136548(m) in increasing order, where A136548(m) = the largest numbers h such that A000203(h) <= k (k = 1,2,3,...), where A000203(h) = sum of divisors of h. - Jaroslav Krizek, Mar 01 2010
Where record values of A022404 occur: A086332(n)=A022404(a(n)). - Reinhard Zumkeller, Jun 21 2010
Positive integers that have no divisors other than 1 and itself (the old definition of prime numbers). - Omar E. Pol, Aug 10 2012
Conjecture: the sequence contains exactly those k such that sigma(k) > k*BigOmega(k). - Irina Gerasimova, Jun 08 2013
Note on the Gerasimova conjecture: all terms in the sequence obviously satisfy the inequality, because sigma(p) = 1+p and BigOmega(p) = 1 for primes p, so 1+p > p*1. For composites, the (opposite) inequality is heuristically correct at least up to k <= 4400000. The general proof requires to show that BigOmega(k) is an upper limit of the abundancy sigma(k)/k for composite k. This proof is easy for semiprimes k=p1*p2 in general, where sigma(k)=1+p1+p2+p1*p2 and BigOmega(k)=2 and p1, p2 <= 2. - R. J. Mathar, Jun 12 2013
Numbers k such that phi(k) + sigma(k) = 2k. - Farideh Firoozbakht, Aug 03 2014
isA008578(n) <=> k is prime to n for all k in {1,2,...,n-1}. - Peter Luschny, Jun 05 2017
In 1751 Leonhard Euler wrote: "Having so established this sign S to indicate the sum of the divisors of the number in front of which it is placed, it is clear that, if p indicates a prime number, the value of Sp will be 1 + p, except for the case where p = 1, because then we have S1 = 1, and not S1 = 1 + 1. From this we see that we must exclude unity from the sequence of prime numbers, so that unity, being the start of whole numbers, it is neither prime nor composite." - Omar E. Pol, Oct 12 2021
a(1) = 1; for n >= 2, a(n) is the least unused number that is coprime to all previous terms. - Jianing Song, May 28 2022
A number p is preprime if p = a*b ==> a = 1 or b = 1. This sequence lists the preprimes in the commutative monoid IN \ {0}. - Peter Luschny, Aug 26 2022
REFERENCES
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 84 at pp. 214-217.
G. Chrystal, Algebra: An Elementary Textbook. Chelsea Publishing Company, 7th edition, (1964), chap. III.7, p. 38.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 11.
H. D. Huskey, Derrick Henry Lehmer [1905-1991]. IEEE Ann. Hist. Comput. 17 (1995), no. 2, 64-68. Math. Rev. 96b:01035
D. H. Lehmer, The sieve problem for all-purpose computers. Math. Tables and Other Aids to Computation, Math. Tables and Other Aids to Computation, 7, (1953). 6-14. Math. Rev. 14:691e
D. N. Lehmer, "List of Prime Numbers from 1 to 10,006,721", Carnegie Institute, Washington, D.C. 1909.
R. F. Lukes, C. D. Patterson and H. C. Williams, Numerical sieving devices: their history and some applications. Nieuw Arch. Wisk. (4) 13 (1995), no. 1, 113-139. Math. Rev. 96m:11082
H. C. Williams and J. O. Shallit, Factoring integers before computers. Mathematics of Computation 1943-1993: a half-century of computational mathematics (Vancouver, BC, 1993), 481-531, Proc. Sympos. Appl. Math., 48, AMS, Providence, RI, 1994. Math. Rev. 95m:11143
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]
Chris K. Caldwell, Angela Reddick, Yeng Xiong, and Wilfrid Keller, The History of the Primality of One: A Selection of Sources, (a dynamic survey), Journal of Integer Sequences, Vol. 15 (2012), #12.9.8.
C. K. Caldwell and Y. Xiong, What is the smallest prime?, arXiv preprint arXiv:1209.2007 [math.HO], 2012, and J. Int. Seq. 15 (2012) #12.9.6
Leonhard Euler, Découverte d’une loi tout extraordinaire des nombres, par rapport à la somme de leurs diviseurs, in Bibliothèque impartiale, 3, 1751, pp. 10-31. Reprinted in Opera Postuma, 1, 1862, p.76-84. Number 175 in the Eneström index.
G. P. Michon, Is 1 a prime number?
Omar E. Pol, Illustration of initial terms
Omar E. Pol, Illustration of initial terms of A008578, A161344, A161345, A161424
PrimeFan, Arguments for and against the primality of 1
A. Reddick and Y. Xiong, The search for one as a prime number: from ancient Greece to modern times, Electronic Journal of Undergraduate Mathematics, Volume 16, 1 - 13, 2012. - From N. J. A. Sloane, Feb 03 2013
J. Todd, Review of Lehmer's tables, Mathematical Tables and Other Aids to Computation, Vol. 11, No. 60, (1957) (on JSTOR.org).
Wikipedia, "Complete" sequence. [Wikipedia calls a sequence "complete" (sic) if every positive integer is a sum of distinct terms. This name is extremely misleading and should be avoided. - N. J. A. Sloane, May 20 2023]
Wikipedia, Dirichlet convolution
FORMULA
a(n) = A000040(n-1).
m is in the sequence iff sigma(m) + phi(m) = A065387(m) = 2m. - Farideh Firoozbakht, Jan 27 2005
a(n) = A158611(n+1) for n >= 1. - Jaroslav Krizek, Jun 19 2009
In the following formulas (based on emails from Jaroslav Krizek and R. J. Mathar), the star denotes a Dirichlet convolution between two sequences, and "This" is A008578.
This = A030014 * A008683. (Dirichlet convolution using offset 1 with A030014)
This = A030013 * A000012. (Dirichlet convolution using offset 1 with A030013)
This = A034773 * A007427. (Dirichlet convolution)
This = A034760 * A023900. (Dirichlet convolution)
This = A034762 * A046692. (Dirichlet convolution)
This * A000012 = A030014. (Dirichlet convolution using offset 1 with A030014)
This * A008683 = A030013. (Dirichlet convolution using offset 1 with A030013)
This * A000005 = A034773. (Dirichlet convolution)
This * A000010 = A034760. (Dirichlet convolution)
This * A000203 = A034762. (Dirichlet convolution)
A002033(a(n))=1. - Juri-Stepan Gerasimov, Sep 27 2009
a(n) = A181363((2*n-1)*2^k), k >= 0. - Reinhard Zumkeller, Oct 16 2010
a(n) = A001747(n)/2. - Omar E. Pol, Jan 30 2012
A060448(a(n)) = 1. - Reinhard Zumkeller, Apr 05 2012
A086971(a(n)) = 0. - Reinhard Zumkeller, Dec 14 2012
Sum_{n>=1} x^a(n) = (Sum_{n>=1} (A002815(n)*x^n))*(1-x)^2. - L. Edson Jeffery, Nov 25 2013
MAPLE
A008578 := n->if n=1 then 1 else ithprime(n-1); fi :
MATHEMATICA
Join[ {1}, Table[ Prime[n], {n, 1, 60} ] ]
NestList[ NextPrime, 1, 57] (* Robert G. Wilson v, Jul 21 2015 *)
oldPrimeQ[n_] := AllTrue[Range[n-1], CoprimeQ[#, n]&];
Select[Range[271], oldPrimeQ] (* Jean-François Alcover, Jun 07 2017, after Peter Luschny *)
PROG
(PARI) is(n)=isprime(n)||n==1
(Magma) [1] cat [n: n in PrimesUpTo(271)]; // Bruno Berselli, Mar 05 2011
(Haskell)
a008578 n = a008578_list !! (n-1)
a008578_list = 1 : a000040_list
-- Reinhard Zumkeller, Nov 09 2011
(Sage)
isA008578 = lambda n: all(gcd(k, n) == 1 for k in (1..n-1))
print([n for n in (1..271) if isA008578(n)]) # Peter Luschny, Jun 07 2017
(GAP)
A008578:=Concatenation([1], Filtered([1..10^5], IsPrime)); # Muniru A Asiru, Sep 07 2017
CROSSREFS
The main entry for this sequence is A000040.
The complement of A002808.
Cf. A000732 (boustrophedon transform).
Cf. A000010, A000203.
Cf. A023626 (self-convolution).
KEYWORD
nonn,easy,nice,changed
AUTHOR
N. J. A. Sloane
STATUS
approved
A033676 Largest divisor of n <= sqrt(n). +10
137
1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 4, 1, 3, 1, 4, 3, 2, 1, 4, 5, 2, 3, 4, 1, 5, 1, 4, 3, 2, 5, 6, 1, 2, 3, 5, 1, 6, 1, 4, 5, 2, 1, 6, 7, 5, 3, 4, 1, 6, 5, 7, 3, 2, 1, 6, 1, 2, 7, 8, 5, 6, 1, 4, 3, 7, 1, 8, 1, 2, 5, 4, 7, 6, 1, 8, 9, 2, 1, 7, 5, 2, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
a(n) = sqrt(n) is a new record if and only if n is a square. - Zak Seidov, Jul 17 2009
a(n) = A060775(n) unless n is a square, when a(n) = A033677(n) = sqrt(n) is strictly larger than A060775(n). It would be nice to have an efficient algorithm to calculate these terms when n has a large number of divisors, as for example in A060776, A060777 and related problems such as A182987. - M. F. Hasler, Sep 20 2011
a(n) = 1 when n = 1 or n is prime. - Alonso del Arte, Nov 25 2012
a(n) is the smallest central divisor of n. Column 1 of A207375. - Omar E. Pol, Feb 26 2019
a(n^4+n^2+1) = n^2-n+1: suppose that n^2-n+k divides n^4+n^2+1 = (n^2-n+k)*(n^2+n-k+2) - (k-1)*(2*n+1-k) for 2 <= k <= 2*n, then (k-1)*(2*n+1-k) >= n^2-n+k, or n^2 - (2*k-1)*n + (k^2-k+1) = (n-k+1/2)^2 + 3/4 < 0, which is impossible. Hence the next smallest divisor of n^4+n^2+1 than n^2-n+1 is at least n^2-n+(2*n+1) = n^2+n+1 > sqrt(n^4+n^2+1). - Jianing Song, Oct 23 2022
REFERENCES
G. Tenenbaum, pp. 268 ff, in: R. L. Graham et al., eds., Mathematics of Paul Erdős I.
LINKS
T. D. Noe, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = n / A033677(n).
a(n) = A161906(n,A038548(n)). - Reinhard Zumkeller, Mar 08 2013
a(n) = A162348(2n-1). - Daniel Forgues, Sep 29 2014
MAPLE
A033676 := proc(n) local a, d; a := 0 ; for d in numtheory[divisors](n) do if d^2 <= n then a := max(a, d) ; end if; end do: a; end proc: # R. J. Mathar, Aug 09 2009
MATHEMATICA
largestDivisorLEQR[n_Integer] := Module[{dvs = Divisors[n]}, dvs[[Ceiling[Length@dvs/2]]]]; largestDivisorLEQR /@ Range[100] (* Borislav Stanimirov, Mar 28 2010 *)
Table[Last[Select[Divisors[n], #<=Sqrt[n]&]], {n, 100}] (* Harvey P. Dale, Mar 17 2017 *)
PROG
(PARI) A033676(n) = {local(d); if(n<2, 1, d=divisors(n); d[(length(d)+1)\2])} \\ Michael B. Porter, Jan 30 2010
(Haskell)
a033676 n = last $ takeWhile (<= a000196 n) $ a027750_row n
-- Reinhard Zumkeller, Jun 04 2012
(Python)
from sympy import divisors
def A033676(n):
d = divisors(n)
return d[(len(d)-1)//2] # Chai Wah Wu, Apr 05 2021
CROSSREFS
Cf. A033677 (n/a(n)), A000196 (sqrt), A027750 (list of divisors), A056737 (n/a(n) - a(n)), A219695 (half of this for odd numbers), A207375 (list the central divisor(s)).
The strictly inferior case is A060775. Cf. also A140271.
Indices of given values: A008578 (1 and the prime numbers: a(n) = 1), A161344 (a(n) = 2), A161345 (a(n) = 3), A161424 (4), A161835 (5), A162526 (6), A162527 (7), A162528 (8), A162529 (9), A162530 (10), A162531 (11), A162532 (12), A282668 (indices of primes).
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane
STATUS
approved
A161344 Numbers k with A033676(k)=2, where A033676 is the largest divisor <= sqrt(k). +10
53
4, 6, 8, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Define a sieve operation with parameter s that eliminates integers of the form s^2 + s*i (i >= 0) from the set A000027 of natural numbers. The sequence lists those natural numbers that are eliminated by the sieve s=2 and cannot be eliminated by any sieve s >= 3. - R. J. Mathar, Jun 24 2009
After a(3)=8 all terms are 2*prime; for n > 3, a(n) = 2*prime(n-1) = 2*A000040(n-1). - Zak Seidov, Jul 18 2009
From Omar E. Pol, Jul 18 2009: (Start)
A classification of the natural numbers A000027.
=============================================================
Numbers k whose largest divisor <= sqrt(k) equals j
=============================================================
j Sequence Comment
=============================================================
1 ..... A008578 1 together with the prime numbers
2 ..... A161344 This sequence
3 ..... A161345
4 ..... A161424
5 ..... A161835
6 ..... A162526
7 ..... A162527
8 ..... A162528
9 ..... A162529
10 .... A162530
11 .... A162531
12 .... A162532
... And so on. (End)
The numbers k whose largest divisor <= sqrt(k) is j are exactly those numbers j*m where m is either a prime >= k or one of the numbers in row j of A163925. - Franklin T. Adams-Watters, Aug 06 2009
See also A163280, the main entry for this sequence. - Omar E. Pol, Oct 24 2009
Also A100484 UNION 8. - Omar E. Pol, Nov 29 2012 (after Seidov and Hasler)
LINKS
Table of n, a(n) for n=1..56.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos
Omar E. Pol, Illustration: Divisors and pi(x)
Omar E. Pol, Illustration of initial terms
Omar E. Pol. Illustration of initial terms of A008578, A161344, A161345, A161424
FORMULA
Equals 2*A000040 union {8}. - M. F. Hasler, Nov 27 2012
a(n) = 2*A046022(n+1) = 2*A175787(n). - Omar E. Pol, Nov 27 2012
MAPLE
isA := proc(n, s) if n mod s <> 0 then RETURN(false); fi; if n/s-s >= 0 then RETURN(true); else RETURN(false); fi; end: isA161344 := proc(n) for s from 3 to n do if isA(n, s) then RETURN(false); fi; od: isA(n, 2) ; end: for n from 1 to 3000 do if isA161344(n) then printf("%d, ", n) ; fi; od; # R. J. Mathar, Jun 24 2009
MATHEMATICA
a[n_] := If[n <= 3, 2n+2, 2*Prime[n-1]]; Table[a[n], {n, 1, 56}] (* Jean-François Alcover, Nov 26 2012, after Zak Seidov *)
PROG
(PARI) a(n)=if(n>3, prime(n-1), n+1)*2 \\ M. F. Hasler, Nov 27 2012
CROSSREFS
Cf. A000005, A018253, A160811, A160812, A161205, A161346, A033676, A008578, A161345, A161424, A161835, A162526, A162527, A162528, A162529, A162530, A162531, A162532, A163925.
Second column of array in A163280. Also, second row of array in A163990.
KEYWORD
easy,nonn
AUTHOR
Omar E. Pol, Jun 20 2009
EXTENSIONS
More terms from R. J. Mathar, Jun 24 2009
Definition added by R. J. Mathar, Jun 28 2009
STATUS
approved
A163280 Square array read by antidiagonals where column k lists the numbers j whose largest divisor <= sqrt(j) is k. +10
30
1, 2, 4, 3, 6, 9, 5, 8, 12, 16, 7, 10, 15, 20, 25, 11, 14, 18, 24, 30, 36, 13, 22, 21, 28, 35, 42, 49, 17, 26, 27, 32, 40, 48, 56, 64, 19, 34, 33, 44, 45, 54, 63, 72, 81, 23, 38, 39, 52, 50, 60, 70, 80, 90, 100, 29, 46, 51, 68, 55, 66, 77, 88, 99, 110, 121, 31, 58, 57, 76, 65, 78, 84, 96, 108, 120, 132, 144 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
This sequence is a permutation of the natural numbers A000027. Note that the first column is formed by 1 together with the prime numbers.
Column k contains exactly those numbers j=k*m where m is either a prime >= j or one of the numbers in row k of A163925. - Franklin T. Adams-Watters, Aug 12 2009
LINKS
Table of n, a(n) for n=1..78.
Omar E. Pol, Illustration of initial terms of column 1: A008578
Omar E. Pol, Illustration of initial terms of column 2: A161344
Omar E. Pol, Illustration of initial terms of columns 1-4: A008578, A161344, A161345, A161424
Index entries for sequences that are permutations of the natural numbers
FORMULA
Column k lists the numbers j such that A033676(j)=k.
EXAMPLE
Array begins:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, ...
2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, ...
3, 8, 15, 24, 35, 48, 63, 80, 99, 120, 143, 168, ...
5, 10, 18, 28, 40, 54, 70, 88, 108, 130, 154, 180, ...
7, 14, 21, 32, 45, 60, 77, 96, 117, 140, 165, 192, ...
11, 22, 27, 44, 50, 66, 84, 104, 126, 150, 176, 204, ...
13, 26, 33, 52, 55, 78, 91, 112, 135, 160, 187, 216, ...
17, 34, 39, 68, 65, 102, 98, 128, 153, 170, 198, 228, ...
19, 38, 51, 76, 75, 114, 105, 136, 162, 190, 209, 264, ...
23, 46, 57, 92, 85, 138, 119, 152, 171, 200, 220, 276, ...
29, 58, 69, 116, 95, 174, 133, 184, 189, 230, 231, 348, ...
31, 62, 87, 124, 115, 186, 147, 232, 207, 250, 242, 372, ...
...
MAPLE
A163280 := proc(n, k) local r, T ; r := 0 ; for T from k^2 by k do if A033676(T) = k then r := r+1 ; if r = n then RETURN(T) ; fi; fi; od: end: # R. J. Mathar, Aug 09 2009
MATHEMATICA
nmax = 12;
pm = Prime[nmax];
sDiv[n_] := Select[Divisors[n], #^2 <= n&][[-1]];
Clear[col]; col[k_] := col[k] = Select[Range[k pm], sDiv[#] == k&];
T[n_, k_ /; 1 <= k <= Length[col[k]]] := col[k][[n]];
Table[T[n-k+1, k], {n, 1, nmax}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 15 2019 *)
CROSSREFS
Rows 1-12: A000290, A002378, A005563, A164004, A100451, A164006, A164007, A164008, A164009, A164010, A164011, A164012.
Columns 1-12: A008578, A161344, A161345, A161424, A161835, A162526, A162527, A162528, A162529, A162530, A162531, A162532.
Another version: A163990.
Cf. A000027, A000040, A033676, A147861, A163100, A164000.
KEYWORD
nonn,tabl
AUTHOR
Omar E. Pol, Aug 07 2009
EXTENSIONS
Edited by R. J. Mathar, Aug 01 2010
Example edited by Jean-François Alcover, Dec 15 2019
STATUS
approved
A161424 Numbers k whose largest divisor <= sqrt(k) equals 4. +10
28
16, 20, 24, 28, 32, 44, 52, 68, 76, 92, 116, 124, 148, 164, 172, 188, 212, 236, 244, 268, 284, 292, 316, 332, 356, 388, 404, 412, 428, 436, 452, 508, 524, 548, 556, 596, 604, 628, 652, 668, 692, 716, 724, 764, 772, 788, 796, 844, 892, 908, 916, 932, 956, 964 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Define a sieve operation with parameter s that eliminates integers of the form s^2 + s*i (i >= 0) from the set A000027 of natural numbers. The sequence lists those natural numbers that are eliminated by the sieve s=4 and cannot be eliminated by any sieve s >= 5. - R. J. Mathar, Jun 24 2009
See A161344 for more information. - Omar E. Pol, Jul 05 2009
See also the array in A163280, the main entry for this sequence. - Omar E. Pol, Oct 24 2009
LINKS
Table of n, a(n) for n=1..54.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos
Omar E. Pol, Illustration: Divisors and pi(x)
Omar E. Pol, Illustration for A008578, A161344, A161345 and A161424 [From Omar E. Pol, Oct 24 2009]
FORMULA
Numbers n such that A033676(n)=4. - Omar E. Pol, Jul 05 2009
MAPLE
isA := proc(n, s) if n mod s <> 0 then RETURN(false); fi; if n/s-s >= 0 then RETURN(true); else RETURN(false); fi; end: isA161424 := proc(n) for s from 5 to n do if isA(n, s) then RETURN(false); fi; od: isA(n, 4) ; end: for n from 1 to 3000 do if isA161424(n) then printf("%d, ", n) ; fi; od; # R. J. Mathar, Jun 24 2009
MATHEMATICA
Select[Range[1, 1000], Function[m, Max[Select[Divisors[m], # <= Sqrt[m] &]] == 4]] (* Ashton Baker, Nov 03 2013 *)
CROSSREFS
Cf. A000005, A018253, A160811, A160812, A161205, A161344, A161345, A161346, A161425, A161428, A033676, A008578, A161835, A162526, A162527, A162528, A162529, A162530, A162531, A162532.
Cf. Fourth column of array in A163280. Also, fourth row of array in A163990. - Omar E. Pol, Oct 24 2009
KEYWORD
easy,nonn
AUTHOR
Omar E. Pol, Jun 20 2009
EXTENSIONS
Terms beyond a(8) from R. J. Mathar, Jun 24 2009
Definition added by R. J. Mathar, Jun 28 2009
STATUS
approved
A161835 Numbers k whose largest divisor <= sqrt(k) is 5. +10
24
25, 30, 35, 40, 45, 50, 55, 65, 75, 85, 95, 115, 125, 145, 155, 185, 205, 215, 235, 265, 295, 305, 335, 355, 365, 395, 415, 445, 485, 505, 515, 535, 545, 565, 635, 655, 685, 695, 745, 755, 785, 815, 835, 865, 895, 905, 955, 965, 985, 995, 1055, 1115, 1135, 1145, 1165, 1195 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
See A161344 for more information. - Omar E. Pol, Jul 05 2009
LINKS
Table of n, a(n) for n=1..56.
FORMULA
Numbers k such that A033676(k)=5. - Omar E. Pol, Jul 05 2009
MATHEMATICA
Select[Range[1, 1000], Function[m, Max[Select[Divisors[m], # <= Sqrt[m] &]] == 4]] (* Ashton Baker, Nov 03 2013 *)
PROG
(PARI) is(n)=divisors(n)[(numdiv(n)+1)\2]==5 \\ - M. F. Hasler, Nov 03 2013
CROSSREFS
Cf. A033676, A008578, A161344, A161345, A161424, A161835, A162526, A162527, A162528, A162529, A162530, A162531, A162532.
KEYWORD
easy,nonn
AUTHOR
Omar E. Pol, Jun 20 2009
EXTENSIONS
Definition and more terms added by R. J. Mathar, Jun 28 2009
STATUS
approved
A162527 Numbers k whose largest divisor <= sqrt(k) equals 7. +10
19
49, 56, 63, 70, 77, 84, 91, 98, 105, 119, 133, 147, 161, 175, 203, 217, 245, 259, 287, 301, 329, 343, 371, 413, 427, 469, 497, 511, 553, 581, 623, 679, 707, 721, 749, 763, 791, 889, 917, 959, 973, 1043, 1057, 1099, 1141, 1169, 1211, 1253, 1267, 1337, 1351 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
See A161344 for more information.
LINKS
Nathaniel Johnston, Table of n, a(n) for n = 1..10000
FORMULA
Numbers k such that A033676(k)=7.
MAPLE
A033676 := proc(n) local dvs; dvs := sort(convert(numtheory[divisors](n), list)) ; op(floor((nops(dvs)+1)/2) , dvs) ; end: for n from 1 to 2000 do if A033676(n) = 7 then printf("%d, ", n) ; fi; od: # R. J. Mathar, Jul 13 2009
MATHEMATICA
ld = 7;
selQ[n_] := AllTrue[Divisors[n], # <= ld || #^2 > n&];
Select[Range[ld, 200] ld, selQ] (* Jean-François Alcover, Apr 14 2020 *)
ld7Q[n_]:=Select[Divisors[n], #<=Sqrt[n]&][[-1]]==7; Select[Range[1400], ld7Q] (* Harvey P. Dale, Jan 13 2023 *)
CROSSREFS
Cf. A033676, A008578, A161344, A161345, A161424, A161835, A162526, A162528, A162529, A162530, A162531, A162532.
KEYWORD
easy,nonn
AUTHOR
Omar E. Pol, Jul 05 2009
EXTENSIONS
More terms from R. J. Mathar, Jul 13 2009
STATUS
approved
A162528 Numbers k whose largest divisor <= sqrt(k) equals 8. +10
19
64, 72, 80, 88, 96, 104, 112, 128, 136, 152, 184, 232, 248, 296, 328, 344, 376, 424, 472, 488, 536, 568, 584, 632, 664, 712, 776, 808, 824, 856, 872, 904, 1016, 1048, 1096, 1112, 1192, 1208, 1256, 1304, 1336, 1384, 1432, 1448, 1528, 1544, 1576, 1592, 1688 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
See A161344 for more information.
LINKS
Table of n, a(n) for n=1..49.
FORMULA
Numbers k such that A033676(k)=8.
MAPLE
A033676 := proc(n) local dvs; dvs := sort(convert(numtheory[divisors](n), list)) ; op(floor((nops(dvs)+1)/2) , dvs) ; end: for n from 1 to 2000 do if A033676(n) = 8 then printf("%d, ", n) ; fi; od: # R. J. Mathar, Jul 13 2009
MATHEMATICA
ld8Q[n_]:=Last[Select[Divisors[n], #<=Sqrt[n]&]]==8; Select[Range[ 2000], ld8Q] (* Harvey P. Dale, Apr 08 2017 *)
CROSSREFS
Cf. A033676, A008578, A161344, A161345, A161424, A161835, A162526, A162527, A162529, A162530, A162531, A162532.
KEYWORD
easy,nonn
AUTHOR
Omar E. Pol, Jul 05 2009
EXTENSIONS
More terms from R. J. Mathar, Jul 13 2009
STATUS
approved
A162530 Numbers k whose largest divisor <= sqrt(k) equals 10. +10
19
100, 110, 120, 130, 140, 150, 160, 170, 190, 200, 230, 250, 290, 310, 370, 410, 430, 470, 530, 590, 610, 670, 710, 730, 790, 830, 890, 970, 1010, 1030, 1070, 1090, 1130, 1270, 1310, 1370, 1390, 1490, 1510, 1570, 1630, 1670, 1730, 1790, 1810, 1910, 1930 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
See A161344 for more information.
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
FORMULA
Numbers k such that A033676(k) = 10.
MAPLE
filter:= n -> andmap(t -> t<=10 or t^2 > n, numtheory:-divisors(n)):
select(filter, [seq(n, n=100..10000, 10)]); # Robert Israel, Aug 16 2018
MATHEMATICA
ld10Q[n_]:=Last[Select[Divisors[n], #<=Sqrt[n]&]]==10; Select[Range[2000], ld10Q] (* Harvey P. Dale, Jan 30 2011 *)
CROSSREFS
Cf. A033676, A008578, A161344, A161345, A161424, A161835, A162526, A162527, A162528, A162529, A162531, A162532.
KEYWORD
easy,nonn
AUTHOR
Omar E. Pol, Jul 05 2009
EXTENSIONS
More terms from Gerard P. Michon, Jul 12 2009
STATUS
approved
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