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

1,1

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)

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 *)

Prog

(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));

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.

Sequence in context: A159294 A332228 A349755 * A066528 A046197 A271730

Adjacent sequences: A387159 A387160 A387161 * A387163 A387164 A387165

Keyword

nonn

Author

Antti Karttunen, Aug 27 2025

Status

approved