A387156 -id:A387156 - cp's OEIS frontend

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

1, 2, 5, 12, 13, 26, 29, 37, 41, 44, 56, 61, 73, 74, 76, 90, 101, 109, 113, 122, 137, 146, 153, 157, 172, 173, 181, 193, 218, 229, 236, 257, 268, 277, 281, 312, 313, 314, 317, 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

Offset: 1

Antti Karttunen, Aug 17 2025

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.

Conjecture 2: A323653 is a subsequence (which would follow from conjecture 1 (c) given there).

Cf. A000203, A003557, A057521, A387156.
Subsequences: A323653 (conjectured), A351549, A386425 (odd composites), A386426 (nondeficient terms).
Cf. also A006872, A351446, A387158.

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 *)

A057521(n)=my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1))
isA386424(n) = (A057521(sigma(n))==A057521(n));

{k | A057521(A000203(k)) = A057521(k)}, or equally, {k | A387156(k) = A003557(k)}.

cp's OEIS Frontend

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

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