%I #48 Jul 28 2019 16:51:54
%S 1,2,2,3,3,6,6,4,4,8,8,12,12,24,24,5,5,10,10,15,15,30,30,20,20,40,40,
%T 60,60,120,120,6,6,12,12,18,18,36,36,24,24,48,48,72,72,144,144,30,30,
%U 60,60,90,90,180,180,120,120,240,240,360,360,720,720,7,7,14,14,21,21
%N If n = 2^k - 1, then a(n) = k+1, otherwise a(n) = (A000523(n)+1)*a(A053645(n)).
%C Each n > 1 occurs 2*A045778(n) times in the sequence.
%C f(n+2^k) = (k+1)*f(n) if 2^k > n+1. - _Robert Israel_, Apr 25 2016
%C If the binary indices of n (row n of A048793) are the positions 1's in its reversed binary expansion, then a(n) is the product of all binary indices of n + 1. The number of binary indices of n is A000120(n), their sum is A029931(n), and their average is A326699(n)/A326700(n). - _Gus Wiseman_, Jul 27 2019
%H Peter Kagey, <a href="/A096111/b096111.txt">Table of n, a(n) for n = 0..10000</a>
%F G.f.: ( prod(k>=1, 1+k*x^(2^(k-1)) )- 1 ) / x. - _Vladeta Jovovic_, Nov 08 2005
%F a(n) is the product of the exponents in the binary expansion of 2*n + 2. - _Peter Kagey_, Apr 24 2016
%p f:= proc(n) local L;
%p L:= convert(2*n+2,base,2);
%p convert(subs(0=NULL,zip(`*`,L, [$0..nops(L)-1])),`*`);
%p end proc:
%p map(f, [$0..100]); # _Robert Israel_, Apr 25 2016
%t CoefficientList[(Product[1 + k x^(2^(k - 1)), {k, 7}] - 1)/x, x] (* _Michael De Vlieger_, Apr 08 2016 *)
%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];Table[Times@@bpe[n+1],{n,0,100}] (* _Gus Wiseman_, Jul 26 2019 *)
%o (Scheme:) (define (A096111 n) (cond ((pow2? (+ n 1)) (+ 2 (A000523 n))) (else (* (+ 1 (A000523 n)) (A096111 (A053645 n))))))
%o (define (pow2? n) (and (> n 0) (zero? (A004198bi n (- n 1)))))
%o (PARI)
%o N=166; q='q+O('q^N);
%o gf= (prod(n=1,1+ceil(log(N)/log(2)), 1+n*q^(2^(n-1)) ) - 1) / q;
%o Vec(gf)
%o /* _Joerg Arndt_, Oct 06 2012 */
%Y Permutation of A096115, i.e. a(n) = A096115(A122198(n+1)) [Note the different starting offsets]. Bisection: A121663. Cf. A096113, A052330.
%Y Cf. A029931.
%Y Cf. A000120, A034797, A048793, A070939, A291166, A326031, A326672, A326673.
%K nonn
%O 0,2
%A _Amarnath Murthy_, Jun 29 2004
%E Edited, extended and Scheme code added by _Antti Karttunen_, Aug 25 2006