A359082 Indices of records in A246600.

1, 3, 15, 63, 255, 495, 4095, 96255, 98175, 130815, 203775, 1048575, 5810175, 6455295, 16777215, 67096575, 88062975, 389656575, 553517055, 850917375, 1157349375, 9141354495, 12826279935, 22828220415, 26818379775, 31684427775, 68719476735, 242870910975, 1168231038975

Offset: 1

Author

Amiram Eldar, Dec 15 2022

Keywords

• nonn
• base

Comments

Numbers k with a record number of divisors d such that the bitwise OR of k and d is equal to k (or equivalently, the bitwise AND of k and d is equal to d).

All the terms are odd since A246600(2*k) = A246600(k).

This sequence is infinite since A246600(2^m-1) = A000005(2^m-1) = A046801(m), and A046801 is unbounded (A046801(2^(m+1)) > A046801(2^m) for all m >= 0).

The corresponding record values are 1, 2, 4, 6, 8, 11, 24, 25, 28, 32, 35, 48, 56, 89, 96, 105, 121, 127, 148, 162, 216, 243, 245, 256, 319, 358, 512, 633, 768, ... .

2*10^11 < a(28) <= 2^48 - 1.

Links

• Martin Ehrenstein, Table of n, a(n) for n = 1..36
• Index entries for sequences related to binary expansion of n.
• Index entries for sequences related to divisors of numbers.

Crossrefs

Cf. A046801, A246600, A359080, A359081, A359083.

Programs

Mathematica

s[n_] := DivisorSum[n, 1 &, BitAnd[n, #] == # &]; seq={}; sm = 0; Do[If[(sn = s[n]) > sm, sm = sn; AppendTo[seq, n]], {n, 1, 10^6}]; seq

PARI

lista(nmax) = {my(list = List(), ndmax = 0, d, s); for(n = 1, nmax, nd = sumdiv(n, d, bitand(d, n)==d); if(nd > ndmax, ndmax = nd; listput(list, n))); Vec(list)};

Extensions

a(28)-a(29) from Martin Ehrenstein, Dec 19 2022