Displaying 1-10 of 24 results found.
Numbers with a record number of divisors that are perfect powers ( A091050).
+20
1
1, 4, 8, 16, 32, 64, 128, 256, 432, 576, 1152, 1296, 1728, 3456, 5184, 10368, 15552, 20736, 41472, 46656, 82944, 129600, 259200, 331776, 373248, 518400, 746496, 1036800, 1166400, 2073600, 3240000, 4665600, 6350400, 12700800, 12960000, 18662400, 25401600, 50803200
EXAMPLE
1 has one divisor that is a perfect power, 1. The next number with more than one perfect power divisor is 4, with 2 such divisors, 1 and 4. Therefore a(2) = 4.
MATHEMATICA
powDivNum[n_] := 1 + DivisorSum[n, 1 &, GCD @@ FactorInteger[#][[;; , 2]] > 1 &]; dm = 0; seq = {}; Do[d = powDivNum[n]; If[d > dm, dm = d; AppendTo[seq, n]], {n, 1, 10^6}]; seq
Maximum exponent in the prime factorization of n.
+10
300
0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 5, 1, 2, 2, 2, 1, 1, 1, 3, 1
COMMENTS
Maximum number of invariant factors among abelian groups of order n. - Álvar Ibeas, Nov 01 2014 a(n) is the highest of the frequencies of the parts of the partition having Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1..r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: a(24) = 3; indeed, the partition having Heinz number 24 = 2*2*2*3 is [1,1,1,2], where the distinct parts 1 and 2 have frequencies 3 and 1, respectively. - Emeric Deutsch, Jun 04 2015 a(n) is the smallest k such that b^(phi(n)+k) == b^k (mod n) for all b.
The Euler phi function can be replaced by the Carmichael lambda function.
Problems:
(*) Are there composite numbers n > 4 such that n == a(n) (mod phi(n))? By Lehmer's totient conjecture, there are no such squarefree numbers.
(**) Are there odd numbers n such that a(n) > 1 and n == a(n) (mod lambda(n))? These are odd numbers n such that a(n) > 1 and b^n == b^a(n) (mod n) for all b.
(***) Are there odd numbers n such that a(n) > 1 and n == a(n) (mod ord_{n}(2))? These are odd numbers n such that a(n) > 1 and 2^n == 2^a(n) (mod n).
Note: if (***) do not exist, then (**) do not exist. (End)
Niven (1969) proved that the asymptotic mean of this sequence is 1 + Sum_{j>=2} 1 - (1/zeta(j)) ( A033150). - Amiram Eldar, Jul 10 2020
FORMULA
Conjecture: a(n) = a( A003557(n)) + 1. This relation together with a(1) = 0 defines the sequence. - Velin Yanev, Sep 02 2017 This conjecture seems very easily provable to me: if the factorization of n is p1^k1 * p2^k2 * ... * pm^km, then the factorization of the largest squarefree divisor of n is p1 * p2 * ... * pm. So the factorization of A003557(n) is p1^(k1-1) * p2^(k2-1) * ... * pm^(km-1) if exponents of zero are allowed, or with the product terms that have an exponent of zero removed if they're not (if that results in an empty product, consider it to be 1 as usual). The formula then follows from the fact that provided all ki >= 1, Max(k1, k2, ..., km) = Max(k1-1, k2-1, ..., km-1) + 1, and Max(k1-1, k2-1, ..., km-1) is not altered by removing the ki-1 values that are 0, provided we treat the empty Max() as being 0. That proves the formula and the provisos about empty products and Max() correspond to a(1) = 0.
Also, for any n, applying the formula Max(k1, k2, ..., km) times to n = p1^k1 * p2^k2 * ... * pm^km reduces all the exponents to zero, i.e., to the case a(1) = 0, so that case and the formula generate the sequence. (End)
Sum_{k=1..n} (-1)^k * a(k) ~ c * n, where c = Sum_{k>=2} 1/((2^k-1)*zeta(k)) = 0.44541445377638761933... . - Amiram Eldar, Jul 28 2024
EXAMPLE
For n = 72 = 2^3*3^2, a(72) = max(exponents) = max(3,2) = 3.
MAPLE
a := 0 ;
for f in ifactors(n)[2] do
a := max(a, op(2, f)) ;
end do:
a ;
# second Maple program:
a:= n-> max(0, seq(i[2], i=ifactors(n)[2])):
MATHEMATICA
Table[If[n == 1, 0, Max @@ Last /@ FactorInteger[n]], {n, 100}] (* Ray Chandler, Jan 24 2006 *)
PROG
(Haskell)
a051903 1 = 0
(Python)
from sympy import factorint
return max(factorint(n).values()) if n > 1 else 0
(Scheme, with memoization-macro definec)
CROSSREFS
Average value is A033150 = 1.7052.... Cf. A002322, A005361, A008479, A028234, A051904, A052409, A067029, A091050, A129132, A327295, A328310, A329885.
Row n of table represents second signature of n: list of exponents >= 2 in canonical prime factorization of n, in nonincreasing order, or 0 if no such exponent exists.
+10
26
0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 3, 2, 0, 3, 2, 0, 0, 0, 5, 0, 0, 0, 2, 2, 0, 0, 0, 3, 0, 0, 0, 2, 2, 0, 0, 4, 2, 2, 0, 2, 0, 3, 0, 3, 0, 0, 0, 2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 2, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0
COMMENTS
The multiset of exponents >=2 in the prime factorization of n completely determines a(n) for over 20 sequences in the database (see crossreferences). It also determines the fractions A034444(n)/ A000005(n) and A037445(n)/ A000005(n). For squarefree numbers, this multiset is { } (the empty multiset). The use of 0 in the table to represent each n with no exponents >=2 in its prime factorization accords with the usual OEIS practice of using 0 to represent nonexistent elements when possible. In comments, the second signature of squarefree numbers will be represented as { }.
For each second signature {S}, there exist values of j and k such that, if the second signature of n is {S}, then A085082(n) is congruent to j modulo k. These values are nontrivial unless {S} = { }. Analogous (but not necessarily identical) values of j and k also exist for each second signature with respect to A088873 and A181796. Each sequence of integers with a given second signature {S} has a positive density, unlike the analogous sequences for prime signatures. The highest of these densities is 6/Pi^2 = 0.607927... for A005117 ({S} = { }).
LINKS
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].
FORMULA
For nonsquarefree n, row n is identical to row A057521(n) of table A212171.
EXAMPLE
First rows of table read: 0; 0; 0; 2; 0; 0; 0; 3; 2; 0; 0; 2;...
12 = 2^2*3 has positive exponents 2 and 1 in its canonical prime factorization (1s are often left implicit as exponents). Since only exponents that are 2 or greater appear in a number's second signature, 12's second signature is {2}.
30 = 2*3*5 has no exponents greater than 1 in its prime factorization. The multiset of its exponents >= 2 is { } (the empty multiset), represented in the table with a 0.
72 = 2^3*3^2 has positive exponents 3 and 2 in its prime factorization, as does 108 = 2^2*3^3. Rows 72 and 108 both read {3,2}.
MATHEMATICA
row[n_] := Select[ FactorInteger[n][[All, 2]], # >= 2 &] /. {} -> 0 /. {k__} -> Sequence[k]; Table[row[n], {n, 1, 100}] (* Jean-François Alcover, Apr 16 2013 *)
PROG
(Magma) &cat[IsEmpty(e)select [0]else Reverse(Sort(e))where e is[pe[2]:pe in Factorisation(n)|pe[2]gt 1]:n in[1..102]]; // Jason Kimberley, Jun 13 2012
CROSSREFS
Functions determined by exponents >=2 in the prime factorization of n:
Multiplicative: A000688, A005361, A008966, A038538, A046951, A049419, A050361, A050377, A056624, A061704, A063775, A162510, A162511, A212181. Sequences that contain all integers of a specific second signature: A005117 (second signature { }), A060687 ({2}), A048109 ({3}).
Number of factorizations of n using perfect powers (elements of A001597) other than 1.
+10
17
1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2
EXAMPLE
The a(1152) = 7 factorizations are (4*4*8*9), (4*8*36), (4*9*32), (8*9*16), (8*144), (9*128), (32*36).
MAPLE
ispp:= proc(n) local F;
F:= ifactors(n)[2];
igcd(op(map(t -> t[2], F)))>1
end proc:
f:= proc(n) local F, np, Q;
F:= map(t -> t[2], ifactors(n)[2]);
np:= mul(ithprime(i)^F[i], i=1..nops(F));
Q:= select(ispp, numtheory:-divisors(np));
G(Q, np)
end proc:
G:= proc(Q, n) option remember; local q, t, k;
if not numtheory:-factorset(n) subset `union`(seq(numtheory:-factorset(q), q=Q)) then return 0 fi;
q:= Q[1]; t:= 0;
for k from 0 while n mod q^k = 0 do
t:= t + procname(Q[2..-1], n/q^k)
od;
t
end proc:
G({}, 1):= 1:
MATHEMATICA
ppQ[n_]:=And[n>1, GCD@@FactorInteger[n][[All, 2]]>1];
facsp[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facsp[n/d], Min@@#>=d&]], {d, Select[Divisors[n], ppQ]}]];
Table[Length[facsp[n]], {n, 100}]
CROSSREFS
Cf. A000961, A001055, A001222, A001597, A001694, A007716, A007916, A045778, A052409, A052410, A052486, A091050, A203025, A303707, A303710.
Number of perfect-power divisors of superprimorials A006939.
+10
13
1, 1, 2, 5, 15, 44, 169, 652, 3106, 15286, 89933, 532476, 3698650, 25749335, 204947216, 1636097441, 14693641859, 132055603656, 1319433514898, 13186485900967, 144978145009105, 1594375302986404, 19128405558986057, 229508085926717076, 2983342885319348522
COMMENTS
A number is a perfect power iff it is 1 or its prime exponents (signature) are not relatively prime.
The n-th superprimorial number is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1).
FORMULA
a(n) = 1 + Sum_{k=2..n} mu(k)*(1 - Product_{i=1..n} 1 + floor(i/k)). - Andrew Howroyd, Aug 30 2020
EXAMPLE
The a(0) = 1 through a(4) = 15 divisors:
1 2 12 360 75600
-------------------------
1 1 1 1 1
4 4 4
8 8
9 9
36 16
25
27
36
100
144
216
225
400
900
3600
MATHEMATICA
chern[n_]:=Product[Prime[i]^(n-i+1), {i, n}];
perpouQ[n_]:=Or[n==1, GCD@@FactorInteger[n][[All, 2]]>1];
Table[Length[Select[Divisors[chern[n]], perpouQ]], {n, 0, 5}]
PROG
(PARI) a(n) = {1 + sum(k=2, n, moebius(k)*(1 - prod(i=1, n, 1 + i\k)))} \\ Andrew Howroyd, Aug 30 2020
CROSSREFS
A336416 gives the same for factorials instead of superprimorials.
A000217 counts prime power divisors of superprimorials.
A006939 gives superprimorials or Chernoff numbers.
A022915 counts permutations of prime indices of superprimorials.
A091050 counts perfect power divisors.
A181818 gives products of superprimorials.
A294068 counts factorizations using perfect powers.
A317829 counts factorizations of superprimorials.
Cf. A000005, A027423, A090630, A118914, A124010, A203025, A251753, A327527, A336419, A336420, A336421, A336426.
Number of ways to write n as a product of a number that is not a perfect power and a squarefree number.
+10
11
0, 1, 1, 1, 1, 3, 1, 0, 1, 3, 1, 3, 1, 3, 3, 0, 1, 3, 1, 3, 3, 3, 1, 2, 1, 3, 0, 3, 1, 7, 1, 0, 3, 3, 3, 3, 1, 3, 3, 2, 1, 7, 1, 3, 3, 3, 1, 2, 1, 3, 3, 3, 1, 2, 3, 2, 3, 3, 1, 7, 1, 3, 3, 0, 3, 7, 1, 3, 3, 7, 1, 3, 1, 3, 3, 3, 3, 7, 1, 2, 0, 3, 1, 7, 3, 3, 3, 2, 1
EXAMPLE
The a(180) = 7 ways are (6*30), (12*15), (18*10), (30*6), (60*3), (90*2), (180*1).
MATHEMATICA
radQ[n_]:=And[n>1, GCD@@FactorInteger[n][[All, 2]]===1];
Table[Length[Select[Divisors[n], radQ[#]&&SquareFreeQ[n/#]&]], {n, 100}]
PROG
(PARI) a(n)={sumdiv(n, d, d<>1 && !ispower(d) && issquarefree(n/d))} \\ Andrew Howroyd, Aug 26 2018
CROSSREFS
Cf. A000961, A001055, A001597, A001694, A005117, A007916, A034444, A091050, A183096, A303386, A303707, A304327, A304328.
a(n) = Sum_{d|n, d = 1 or not a perfect power} mu(n/d).
+10
11
1, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, -1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 1, 1, -1, 0, 0, 0, 0, 0
COMMENTS
The Moebius function mu is defined by mu(n) = (-1)^k if n is a product of k distinct primes and mu(n) = 0 otherwise.
Up to n = 10^7 this sequence only takes values in {-2, -1, 0, 1, 2}. Is this true in general?
MATHEMATICA
Table[Sum[If[GCD@@FactorInteger[d][[All, 2]]===1, MoebiusMu[n/d], 0], {d, Divisors[n]}], {n, 100}]
CROSSREFS
Cf. A000005, A000961, A001221, A001597, A001694, A005117, A007916, A008683, A091050, A203025, A304326, A304327, A304364, A304365, A304369.
Number of ways to write n as a product of a perfect power and a squarefree number.
+10
10
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1
COMMENTS
First term greater than 2 is a(746496) = 3.
EXAMPLE
The a(746496) = 3 ways are 12^5*3, 72^3*2, 864^2*1.
MATHEMATICA
Table[Length[Select[Divisors[n], (#===1||GCD@@FactorInteger[#][[All, 2]]>1)&&SquareFreeQ[n/#]&]], {n, 100}]
CROSSREFS
Cf. A000961, A001055, A001597, A005117, A007916, A034444, A091050, A183096, A203025, A246549, A303386, A304326, A304328.
a(n) = number of divisors of n that are not perfect powers.
+10
9
0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 3, 1, 1, 4, 1, 4, 3, 3, 1, 5, 1, 3, 1, 4, 1, 7, 1, 1, 3, 3, 3, 5, 1, 3, 3, 5, 1, 7, 1, 4, 4, 3, 1, 6, 1, 4, 3, 4, 1, 5, 3, 5, 3, 3, 1, 10, 1, 3, 4, 1, 3, 7, 1, 4, 3, 7, 1, 7, 1, 3, 4, 4, 3, 7, 1, 6, 1, 3, 1, 10, 3, 3, 3, 5, 1, 10, 3, 4, 3, 3, 3, 7, 1, 4, 4, 5
FORMULA
a(1) = 0, a(p) = 1, a(pq) = 3, a(pq...z) = 2^k - 1, a(p^k) = 1, for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.
EXAMPLE
For n = 12, set of such divisors is {2, 3, 6, 12}; a(12) = 4.
The "rootless" zeta function. Dirichlet inverse of the function defined by r(n) = (-1)^Omega(n) if n is 1 or not a perfect power and r(n) = 0 otherwise.
+10
9
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 7, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 3, 1, 2, 2, 5, 1, 1, 1, 2, 1
COMMENTS
Omega(n) = A001222(n) is the number of prime factors of n counted with multiplicity. First occurrence of k: 1, 12, 48, 60, 36, 3072, 72, 420, 240, 786432, 3145728, 144, 216, ..., . - Robert G. Wilson v, Jul 22 2018 Records: 1, 2, 5, 7, 12, 13, 15, 18, 26, 37, 38, 57, 60, 67, 81, 96, 142, 165, 199, 221, 234, ..., . - Robert G. Wilson v, Jul 22 2018
FORMULA
a(1) = 1 and a(n > 1) = -Sum_{d|n, d not a perfect power} (-1)^Omega(d) * a(n/d).
MATHEMATICA
a[n_]:=a[n]=If[n==1, 1, -Sum[(-1)^PrimeOmega[d]*a[n/d], {d, Select[Rest[Divisors[n]], GCD@@FactorInteger[#][[All, 2]]==1&]}]];
Array[a, 100]
CROSSREFS
Positions of entries greater than 1 appear to be A126706. Cf. A000005, A000012, A000961, A001221, A001222, A001597, A005117, A007916, A008683, A008966, A091050, A303554, A304326, A304362, A304819.
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