cp's OEIS Frontend

A065413 Number of positive solutions to "numbers that are n times their number of binary 1's".

%I A065413 #20 May 11 2016 16:23:31
%S A065413 1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,0,1,1,1,3,1,0,0,2,1,1,1,1,1,1,1,
%T A065413 0,1,2,0,1,1,2,1,1,1,1,3,0,1,0,0,1,0,2,2,0,1,2,1,1,1,1,1,1,1,1,1,0,1,
%U A065413 2,0,2,1,1,2,1,0,2,1,2,1,2,2,2,1,2,1,1,1,2,1,2,3,1,0,0,1,0,0,1,0,2,1,0,0,2
%N A065413 Number of positive solutions to "numbers that are n times their number of binary 1's".
%C A065413 Equivalently, this is the number of ways to write n as an arithmetic mean of distinct powers of 2. [_Brian Kell_, Mar 01 2009]
%H A065413 Peter Kagey, <a href="/A065413/b065413.txt">Table of n, a(n) for n = 1..10000</a>
%e A065413 a(23)=3 since 69, 92 and 115 are written in binary as 1000101, 1011100 and 1110011 and 69=23*3, 92=23*4 and 115=23*5.
%p A065413 N:= 1000: # to get a(1) to a(N)
%p A065413 A:= Vector(N):
%p A065413 for x from 1 while x/(1+ilog2(x)) <= N do
%p A065413   v:= x/convert(convert(x,base,2),`+`);
%p A065413   if v::integer and v <= N then
%p A065413     A[v]:= A[v]+1
%p A065413   fi
%p A065413 od:
%p A065413 seq(A[i],i=1..N); # _Robert Israel_, May 06 2016
%p A065413 # alternative program
%p A065413 read("transforms") :
%p A065413 A065413 := proc(n)
%p A065413     local bdgs,a,x;
%p A065413     a := 0 ;
%p A065413     for bdgs from 1 do
%p A065413         x := n*bdgs ;
%p A065413         # x must have bdgs bits set, so x =bdgs*n >= 2^bdgs-1.
%p A065413         if n < (2^bdgs-1)/x then
%p A065413             break;
%p A065413         elif wt(x) = bdgs then
%p A065413             a := a+1 ;
%p A065413         end if;
%p A065413     end do:
%p A065413     a ;
%p A065413 end proc: # _R. J. Mathar_, May 11 2016
%Y A065413 Cf. A000120, A037478, A058898, A272797 (greedy inverse).
%K A065413 base,nonn
%O A065413 1,23
%A A065413 _Henry Bottomley_, Nov 23 2001