A380848 Numbers k such that A380845(k) = 4*k.

123832800, 247695840, 268337160, 495421920, 536707080, 990874080, 1073446920, 1981778400, 2146926600, 3963587040, 4293885960, 7927204320, 8587804680, 15854438880, 17175642120, 31708908000, 34351317000, 63417846240, 68702666760, 124884879840, 126713795040, 126835722720

Offset: 1

Author

Amiram Eldar, Feb 05 2025

Keywords

• nonn
• base

Comments

Analogous to 4-perfect numbers (A027687) with A380845 instead of A000203.

All the terms are 4-abundant numbers (A068404), because A380845(k) <= A000203(k) with equality only when k is a power of 2, and powers of 2 are deficient numbers (A005100).

Are there numbers k such that A380845(k) = m*k for integers m >= 5? There are none below 1.6*10^11.

Crossrefs

Cf. A000203, A005100, A027687, A380845, A380846, A380847.

Subsequence of A068404.

Programs

Mathematica

q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] == 4*k]; Select[Range[3*10^8], q]

PARI

isok(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) == 4*k;}

Extensions

a(19)-a(22) from Jinyuan Wang, Feb 12 2025