A381069 Numbers k that have a record number of divisors that have the same binary weight as k.

1, 2, 4, 8, 16, 32, 64, 72, 144, 288, 576, 1080, 2160, 4320, 8640, 17280, 34560, 69120, 99360, 136080, 198720, 272160, 397440, 529200, 544320, 1058400, 2116800, 3160080, 4233600, 6320160, 8467200, 12640320, 16934400, 25280640, 50561280, 76744800, 101122560, 102816000

Offset: 1

Author

Amiram Eldar, Feb 12 2025

Keywords

• nonn
• base

Comments

Indices of records of A380844, i.e., numbers k such that A380844(k) > A380844(m) for all m < k.

This sequence is infinite since A380844 is unbounded (e.g., A380844(2^n) = n+1).

Analogous to highly composite numbers (A002182) with the number of divisors with the same binary weight (A380844) instead of the number of divisors (A000005).

The corresponding record values are 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 20, 24, 28, ... (see the link for more values).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..60
• Amiram Eldar, Table of n, a(n), A380844(a(n)) for n = 1..60

Crossrefs

Cf. A000005, A002182, A380844, A381070.

Programs

Mathematica

seq[lim_] := Module[{h, d, dmax = 0, s = {}}, Do[h = DigitCount[k, 2, 1]; d = DivisorSum[k, 1 &, DigitCount[#, 2, 1] == h &]; If[d > dmax, dmax = d; AppendTo[s, k]], {k, 1, lim}]; s]; seq[10^5]

PARI

list(lim) = {my(h, d, dmax = 0); for(k = 1, lim, h = hammingweight(k); d = sumdiv(k, d, hammingweight(d) == h); if(d > dmax, dmax = d; print1(k, ", ")));}