A387162 Numbers k satisfying Euler's criterion for odd perfect numbers (A228058), such that sigma(k)+k is also a multiple of 3, and sigma(k) preserves the 3-adic valuation of k, where sigma is the sum of divisors function.

153, 325, 801, 925, 1525, 1573, 1773, 1825, 2097, 2205, 2425, 2725, 3757, 3825, 3925, 4041, 4477, 4525, 4689, 4825, 5013, 5725, 6025, 6877, 6925, 6957, 7381, 7605, 7825, 7929, 8125, 8425, 8577, 8725, 8833, 9325, 9549, 9873, 9925, 10225, 10525, 10693, 10825, 10933, 11425, 11493, 11737, 12789, 13189, 13437, 13525

Offset: 1

Author

Antti Karttunen, Aug 27 2025

Keywords

• nonn
• new

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10001
• Index entries for sequences where odd perfect numbers must occur, if they exist at all
• Index entries for sequences related to sigma(n)

Crossrefs

Intersection of A228058 and A349752.

Subsequence of A349755 from which this differs for the first time at n=109, with a(109) = 31225, while A349755(109) = 31213.

Probably the intersection of A349755 and A386429.

Programs

Mathematica

nn=275;n=1;a228058={};While[Length[a228058 ] < nn,n=n+2;{p,e}=Transpose[FactorInteger[n]];od=Select[e,OddQ];If[Length[e]>1&&Length[od]==1&&Mod[od[[1]], 4]==1&&Mod[p[[Position[e, od[[1]]][[1,1]]]],4]==1,AppendTo[a228058,n]]];lim=a228058[[-1]];a349752=Select[Range[1,lim,2],Divisible[(s=DivisorSigma[1,#])+#,3] && IntegerExponent[s,3]==IntegerExponent[#,3]&];Intersection[a228058,a349752] (* James C. McMahon, Aug 27 2025 *)

PARI

isA228058(n) = if(!(n%2)||(omega(n)<2), 0, my(f=factor(n), y=0); for(i=1, #f~, if(1==(f[i, 2]%4), if((1==y)||(1!=(f[i, 1]%4)), return(0), y=1), if(f[i, 2]%2, return(0)))); (y));
isA349752(n) = if(!(n%2), 0, my(s=sigma(n)); (0==(s+n)%3) && valuation(s, 3)==valuation(n, 3));
isA387162(n) = (isA349752(n) && isA228058(n));