Search: a161835 -id:a161835
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5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 19, 23, 25, 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
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OFFSET
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1,1
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LINKS
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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A033676
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Largest divisor of n <= sqrt(n).
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+10
137
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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
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OFFSET
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1,4
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COMMENTS
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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) 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
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REFERENCES
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G. Tenenbaum, pp. 268 ff, in: R. L. Graham et al., eds., Mathematics of Paul Erdős I.
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LINKS
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FORMULA
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(Haskell)
a033676 n = last $ takeWhile (<= a000196 n) $ a027750_row n
(Python)
from sympy import divisors
d = divisors(n)
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CROSSREFS
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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).
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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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
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OFFSET
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1,1
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COMMENTS
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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
A classification of the natural numbers A000027.
=============================================================
Numbers k whose largest divisor <= sqrt(k) equals j
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j Sequence Comment
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1 ..... A008578 1 together with the prime numbers
... 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
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LINKS
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FORMULA
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MAPLE
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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
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MATHEMATICA
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PROG
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(PARI) a(n)=if(n>3, prime(n-1), n+1)*2 \\ M. F. Hasler, Nov 27 2012
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CROSSREFS
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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.
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A161345
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Numbers k whose largest divisor <= sqrt(k) is 3.
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+10
39
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9, 12, 15, 18, 21, 27, 33, 39, 51, 57, 69, 87, 93, 111, 123, 129, 141, 159, 177, 183, 201, 213, 219, 237, 249, 267, 291, 303, 309, 321, 327, 339, 381, 393, 411, 417, 447, 453, 471, 489, 501, 519, 537, 543, 573, 579, 591, 597, 633, 669, 681, 687, 699, 717, 723
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OFFSET
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1,1
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COMMENTS
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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=3 and cannot be eliminated by any sieve s >= 4. - R. J. Mathar, Jun 24 2009
See also the array in A163280, the main entry for this sequence. - Omar E. Pol, Oct 24 2009
Union of {12, 18, 27} and all the numbers of the form 3*p, where p is an odd prime. - Amiram Eldar, Apr 17 2024
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LINKS
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FORMULA
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MAPLE
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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: isA161345 := proc(n) for s from 4 to n do if isA(n, s) then RETURN(false); fi; od: isA(n, 3) ; end: for n from 1 to 3000 do if isA161345(n) then printf("%d, ", n) ; fi; od; # R. J. Mathar, Jun 24 2009
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MATHEMATICA
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md3Q[n_]:=Max[Select[Divisors[n], #<=Sqrt[n]&]]==3; Select[Range[800], md3Q] (* Harvey P. Dale, Aug 12 2013 *)
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CROSSREFS
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Cf. A000005, A018253, A160811, A160812, A161205, A161344, A161346, A033676, A008578, A161424, A161835, A162526, A162527, A162528, A162529, A162530, A162531, A162532.
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A163280
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Square array read by antidiagonals where column k lists the numbers j whose largest divisor <= sqrt(j) is k.
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+10
30
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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FORMULA
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Column k lists the numbers j such that A033676(j)=k.
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EXAMPLE
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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, ...
...
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MAPLE
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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
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MATHEMATICA
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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]];
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CROSSREFS
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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.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A161424
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Numbers k whose largest divisor <= sqrt(k) equals 4.
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+10
28
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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
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OFFSET
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1,1
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COMMENTS
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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 also the array in A163280, the main entry for this sequence. - Omar E. Pol, Oct 24 2009
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LINKS
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FORMULA
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MAPLE
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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
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MATHEMATICA
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Select[Range[1, 1000], Function[m, Max[Select[Divisors[m], # <= Sqrt[m] &]] == 4]] (* Ashton Baker, Nov 03 2013 *)
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CROSSREFS
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Cf. A000005, A018253, A160811, A160812, A161205, A161344, A161345, A161346, A161425, A161428, A033676, A008578, A161835, A162526, A162527, A162528, A162529, A162530, A162531, A162532.
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A162527
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Numbers k whose largest divisor <= sqrt(k) equals 7.
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+10
19
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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
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OFFSET
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1,1
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COMMENTS
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LINKS
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FORMULA
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MAPLE
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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
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MATHEMATICA
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ld = 7;
selQ[n_] := AllTrue[Divisors[n], # <= ld || #^2 > n&];
ld7Q[n_]:=Select[Divisors[n], #<=Sqrt[n]&][[-1]]==7; Select[Range[1400], ld7Q] (* Harvey P. Dale, Jan 13 2023 *)
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CROSSREFS
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Cf. A033676, A008578, A161344, A161345, A161424, A161835, A162526, A162528, A162529, A162530, A162531, A162532.
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A162528
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Numbers k whose largest divisor <= sqrt(k) equals 8.
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+10
19
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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
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OFFSET
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1,1
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COMMENTS
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LINKS
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FORMULA
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MAPLE
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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
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MATHEMATICA
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ld8Q[n_]:=Last[Select[Divisors[n], #<=Sqrt[n]&]]==8; Select[Range[ 2000], ld8Q] (* Harvey P. Dale, Apr 08 2017 *)
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CROSSREFS
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Cf. A033676, A008578, A161344, A161345, A161424, A161835, A162526, A162527, A162529, A162530, A162531, A162532.
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A162530
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Numbers k whose largest divisor <= sqrt(k) equals 10.
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+10
19
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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
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OFFSET
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1,1
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COMMENTS
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LINKS
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FORMULA
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Numbers k such that A033676(k) = 10.
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MAPLE
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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
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MATHEMATICA
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ld10Q[n_]:=Last[Select[Divisors[n], #<=Sqrt[n]&]]==10; Select[Range[2000], ld10Q] (* Harvey P. Dale, Jan 30 2011 *)
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CROSSREFS
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Cf. A033676, A008578, A161344, A161345, A161424, A161835, A162526, A162527, A162528, A162529, A162531, A162532.
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A162532
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Numbers k whose largest divisor <= sqrt(k) equals 12.
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+10
19
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144, 156, 168, 180, 192, 204, 216, 228, 264, 276, 348, 372, 444, 492, 516, 564, 636, 708, 732, 804, 852, 876, 948, 996, 1068, 1164, 1212, 1236, 1284, 1308, 1356, 1524, 1572, 1644, 1668, 1788, 1812, 1884, 1956, 2004, 2076, 2148, 2172, 2292, 2316, 2364
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OFFSET
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1,1
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COMMENTS
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LINKS
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FORMULA
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MAPLE
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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 3500 do if A033676(n) = 12 then printf("%d, ", n) ; fi; od: # R. J. Mathar, Jul 13 2009
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MATHEMATICA
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ld12Q[n_]:=First[Select[Reverse[Divisors[n]], #<=Sqrt[n]&]]==12; Select[ 12*Range[ 200], ld12Q] (* Harvey P. Dale, Mar 29 2013 *)
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CROSSREFS
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Cf. A033676, A008578, A161344, A161345, A161424, A161835, A162526, A162527, A162528, A162529, A162530, A162531.
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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