A386424 Numbers k such that sigma(k) has the same powerful part as k has, 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

1,2

Comments

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).

Conjecture 3: The only common term with A387727 is 1. - Antti Karttunen, Sep 12 2025

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences related to sigma(n).

Index entries for sequences where odd perfect numbers must occur, if they exist at all.

Formula

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

Mathematica

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 *)

Prog

(PARI)
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));

Crossrefs

Cf. A000203, A003557, A057521, A387156.

Subsequences: A323653 (conjectured), A351549, A386425 (odd composites), A386426 (nondeficient terms).

Cf. also A006872, A351446, A387158, A387727.

Sequence in context: A191368 A392292 A392293 * A085227 A324601 A305310

Adjacent sequences: A386421 A386422 A386423 * A386425 A386426 A386427

Keyword

nonn

Author

Antti Karttunen, Aug 17 2025

Status

approved