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

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

Author

Antti Karttunen, Feb 16 2022

Keywords

• nonn

Examples

			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.
		

Links

• Index entries for sequences related to sigma(n)

Crossrefs

Positions of nonzero terms in A351539.

Cf. A000203, A351541 (subsequence).

Probably subsequence: A007691 \ (A323653 U A336702).

Cf. also A336353.

Programs

Mathematica

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 *)

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