A359083 - OEIS

%I #6 Dec 17 2022 08:26:40

%S 1,3,15,63,255,891,4095,262143,1048575,16777215,68719476735

%N Numbers k such that A246600(k) = A000005(k) and A000005(k) sets a new record.

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

%C All the terms are odd since all the terms of A359080 are odd.

%C The corresponding numbers of divisors are 1, 2, 4, 6, 8, 10, 24, 32, 48, 96, 512, ... .

%C a(12) > 3*10^11, if it exists.

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>.

%H <a href="/index/Di#divisors">Index entries for sequences related to divisors of numbers</a>.

%t s[n_] := DivisorSum[n, 1 &, BitAnd[n, #] == # &]; seq={}; dm = 0; Do[d = DivisorSigma[0, n]; If[d > dm && d == s[n], dm = d; AppendTo[seq, n]], {n, 1, 2*10^7}]; seq

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

%Y Subsequence of A359080.

%Y Cf. A000005, A246600, A359081, A359082.

%K nonn,base,more

%O 1,2

%A _Amiram Eldar_, Dec 15 2022