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A386425 Odd composites k such that sigma(k) has the same powerful part as k, where sigma is the sum of divisors function.

%I A386425 #22 Aug 18 2025 09:34:37
%S A386425 153,801,1773,3725,4689,4753,5013,6957,8577,8725,9549,9873,11493,
%T A386425 13437,14409,15381,18621,19269,21213,21537,23481,25101,26073,26225,
%U A386425 28989,29161,29313,29961,32229,33849,34173,36117,38061,39033,40653,42597,43893,47457,47781,48725,48753,51669,52317,54261,56953,57177,57501
%N A386425 Odd composites k such that sigma(k) has the same powerful part as k, where sigma is the sum of divisors function.
%C A386425 By definition, the sequence contains all odd perfect numbers, and also includes any hypothetical odd triperfect number that is not a multiple of 3 (see A005820 and A347391), and similarly, any odd term of A046060 that is not a multiple of 5, etc. If there are no squares in this sequence (see conjecture in A386424), then the latter categories of numbers certainly do not exist, and this is then a subsequence of A228058.
%C A386425 The first nondeficient term is a(32315) = 81022725. See A386426.
%H A386425 Antti Karttunen, <a href="/A386425/b386425.txt">Table of n, a(n) for n = 1..25000</a>
%H A386425 <a href="/index/O#opnseqs">Index entries for sequences where odd perfect numbers must occur, if they exist at all</a>
%H A386425 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A386425 {k | k is odd composite and A003557(A000203(k)) = A003557(k)}.
%t A386425 rad[n_] := Times @@ First /@ FactorInteger[n];a057521[n_] := n/Denominator[n/rad[n]^2];Select[Range[9,57501,2],!PrimeQ[#]&&a057521[DivisorSigma[1,#]]==a057521[#]&] (* _James C. McMahon_, Aug 18 2025 *)
%o A386425 (PARI)
%o A386425 A057521(n)=my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1))
%o A386425 isA386425(n) = ((n>1) && (n%2) && !isprime(n) && (A057521(sigma(n))==A057521(n)));
%Y A386425 Intersection of A071904 and A386424.
%Y A386425 Nonsquare terms form a subsequence of A228058.
%Y A386425 Cf. A000203, A003557, A057521, A386426 (nondeficient terms).
%Y A386425 Cf. also A324647, A349749.
%K A386425 nonn
%O A386425 1,1
%A A386425 _Antti Karttunen_, Aug 17 2025