A351539 -id:A351539 - cp's OEIS frontend

A351439 Number of prime factors p of n such that p^(1+valuation(n,p)) divides sigma(n).

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1

Offset: 1

Antti Karttunen, Feb 16 2022

nonn

			For n = 30 = 2*3*5, sigma(30) = 72 = 2^3 * 3^2 and thus for two of the three prime factors of 30, a higher power of the same prime divides sigma(30), and therefore a(30) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A001221, A007691, A286561, A351539.

Cf. also A336352.

{0}~Join~Table[Function[m, Count[FactorInteger[n][[All, 1]], ?(Mod[m, #^(1 + IntegerExponent[n, #])] == 0 &)]][DivisorSigma[1, n]], {n, 2, 108}] (* _Michael De Vlieger, Feb 16 2022 *)

A351439(n) = { my(f=factor(n),s=sigma(n)); sum(k=1,#f~,(0==(s%(f[k,1]^(1+f[k,2]))))); };

For all n >= 1, a(n) <= A001221(n). [Apparently this is sharp for n > 1].

For all n >= 1, a(n) >= A351539(n).

A351540 Numbers k that have an odd prime factor p such that p^(1+valuation(k,p)) divides sigma(k).

Original entry on oeis.org

30, 51, 66, 96, 102, 120, 138, 159, 165, 174, 204, 210, 213, 246, 255, 264, 267, 282, 294, 306, 318, 321, 330, 345, 354, 357, 364, 390, 408, 426, 435, 462, 477, 480, 498, 510, 534, 537, 552, 561, 570, 591, 606, 615, 636, 642, 660, 663, 672, 678, 679, 690, 696, 699, 705, 714, 735, 745, 750, 753, 759, 760, 765, 786

Offset: 1

Antti Karttunen, Feb 16 2022

nonn

			30 = 2 * 3 * 5 is present as sigma(30) = 72 = 2^3 * 3^2, and thus there is at least one odd prime factor (in this case 3) such that a higher power of the same prime divides the sum of divisors of the same number.

Index entries for sequences related to sigma(n)

Positions of nonzero terms in A351539.
Cf. A000203, A351541 (subsequence).
Probably subsequence: A007691 \ (A323653 U A336702).
Cf. also A336353.

Select[Range[2, 800], Function[{k, s}, AnyTrue[DeleteCases[FactorInteger[k][[All, 1]], 2], Mod[s, #^(1 + IntegerExponent[k, #])] == 0 &]] @@ {#, DivisorSigma[1, #]} &] (* Michael De Vlieger, Feb 16 2022 *)

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]))))); };
isA351540(n) = (A351539(n)>0);

cp's OEIS Frontend

A351439 Number of prime factors p of n such that p^(1+valuation(n,p)) divides sigma(n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula