A387727 Numbers k such that the powerful part of the sum of divisors of k (A387726) is greater than or equal to k.

1, 3, 7, 21, 22, 30, 31, 46, 51, 55, 66, 70, 71, 81, 85, 93, 94, 102, 107, 110, 115, 119, 127, 138, 142, 154, 156, 159, 165, 170, 187, 199, 210, 213, 214, 217, 230, 235, 238, 253, 255, 265, 282, 291, 310, 318, 321, 322, 330, 343, 345, 355, 357, 364, 371, 374, 381, 382, 385, 391, 400, 416, 426, 431, 462, 470, 472

Offset

1,2

Comments

If a hypothetical odd perfect number is in A386428, then it must also reside in this sequence (and in subsequence A387729), and naturally also in A386424.

Links

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

Index entries for sequences related to sigma(n).

Example

21 is present as sigma(21) = 32 = 2^5, whose powerful part is 32, which is >= 21.
416 is present as sigma(416) = 882 = 2 * 3^2 * 7^2, whose powerful part is 3^2 * 7^2 = 441, which is >= 416.

Mathematica

A387727Q[k_] := Times @@ Power @@@ Select[FactorInteger[DivisorSigma[1, k]], Last[#] > 1 &] >= k;
Select[Range[500], A387727Q] (* Paolo Xausa, Sep 13 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)); };
A387726(n) = A057521(sigma(n));
is_A387727(n) = (A387726(n) >= n);

Crossrefs

Cf. A000203, A001694, A057521, A387726, A387728 (complement).

Disjoint union of A180097 and A387729.

Differs from A180097 first at n=62, where a(62) = 416, while A180097(62) = 426.

Cf. A386424, A386428.

Sequence in context: A174348 A196035 A196038 * A180097 A065496 A322922

Adjacent sequences: A387724 A387725 A387726 * A387728 A387729 A387730

Keyword

nonn

Author

Antti Karttunen, Sep 12 2025

Status

approved