A351539 Number of odd prime factors p of n such that p^(1+valuation(n,p)) divides sigma(n).

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 1, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset

1,2235

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences related to sigma(n)

Formula

For all n >= 1, a(n) <= A351439(n) <= A001221(n).

Example

For n = 2235 = 3 * 5 * 149, sigma(2235) = 3600 = 2^4 * 3^2 * 5^2, and thus for two of the three odd prime factors of 2235, a higher power of the same prime divides sigma(2235), and therefore a(2235) = 2. This is also the least n for which a(n) > 1.

Mathematica

{0}~Join~Table[Function[m, Count[DeleteCases[FactorInteger[n][[All, 1]], 2], _?(Mod[m, #^(1 + IntegerExponent[n, #])] == 0 &)]][DivisorSigma[1, n]], {n, 2, 120}] (* Michael De Vlieger, Feb 16 2022 *)

Prog

(PARI) A351539(n) = { my(f=factor(n), s=sigma(n)); sum(k=1, #f~, (f[k, 1]%2)&&(0==(s%(f[k, 1]^(1+f[k, 2]))))); };

Crossrefs

Cf. A000203, A001221, A286561, A351439, A351540 (positions of nonzero terms).

Sequence in context: A011744 A358754 A354996 * A354926 A011743 A011742

Adjacent sequences: A351536 A351537 A351538 * A351540 A351541 A351542

Keyword

nonn

Author

Antti Karttunen, Feb 16 2022

Status

approved