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A386424 Numbers k such that sigma(k) has the same powerful part as k, where sigma is the sum of divisors function.

%I A386424 #24 Aug 21 2025 14:39:15
%S A386424 1,2,5,12,13,26,29,37,41,44,56,61,73,74,76,90,101,109,113,122,137,146,
%T A386424 153,157,172,173,181,193,218,229,236,257,268,277,281,312,313,314,317,
%U A386424 353,362,373,386,389,397,401,409,421,433,457,458,461,509,522,524,528,541,554,560,569,601,613,617,626,641,652,653
%N A386424 Numbers k such that sigma(k) has the same powerful part as k, where sigma is the sum of divisors function.
%C A386424 Conjecture 1: the initial 1 is the only square in this sequence, and a(2) = 2 is the only term that is twice a square.
%C A386424 Conjecture 2: A323653 is a subsequence (which would follow from conjecture 1 (c) given there).
%H A386424 Antti Karttunen, <a href="/A386424/b386424.txt">Table of n, a(n) for n = 1..10000</a>
%H A386424 <a href="/index/O#opnseqs">Index entries for sequences where odd perfect numbers must occur, if they exist at all</a>
%H A386424 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A386424 {k | A057521(A000203(k)) = A057521(k)}, or equally, {k | A387156(k) = A003557(k)}.
%t A386424 rad[n_] := Times @@ First /@ FactorInteger[n];a057521[n_] := n/Denominator[n/rad[n]^2];Select[Range[653],a057521[DivisorSigma[1,#]]==a057521[#]&] (* _James C. McMahon_, Aug 18 2025 *)
%o A386424 (PARI)
%o A386424 A057521(n)=my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1))
%o A386424 isA386424(n) = (A057521(sigma(n))==A057521(n));
%Y A386424 Cf. A000203, A003557, A057521, A387156.
%Y A386424 Subsequences: A323653 (conjectured), A351549, A386425 (odd composites), A386426 (nondeficient terms).
%Y A386424 Cf. also A006872, A351446, A387158.
%K A386424 nonn,new
%O A386424 1,2
%A A386424 _Antti Karttunen_, Aug 17 2025