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

1,2

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

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]

Prog

(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, ", "))); }

Crossrefs

Cf. A000005, A002182, A380844, A381070.

Sequence in context: A036131 A115424 A372944 * A225878 A323097 A272985

Adjacent sequences: A381066 A381067 A381068 * A381070 A381071 A381072

Keyword

nonn,base,new

Author

Amiram Eldar, Feb 12 2025

Status

approved