login
For any nonnegative integer n with binary expansion Sum_{k = 1..w} 2^e_k, let m be the least integer such that the values e_k mod m are all distinct; a(n) = Sum_{k = 1..w} 2^(e_k mod m).
1

%I #10 Oct 17 2022 08:37:24

%S 0,1,1,3,1,5,3,7,1,3,3,11,3,13,7,15,1,3,3,19,6,7,7,23,3,25,11,27,7,29,

%T 15,31,1,3,6,7,3,7,7,39,5,11,7,43,14,15,15,47,3,7,19,51,7,53,23,55,7,

%U 57,27,59,15,61,31,63,1,5,3,7,5,7,7,71,3,13,14,15

%N For any nonnegative integer n with binary expansion Sum_{k = 1..w} 2^e_k, let m be the least integer such that the values e_k mod m are all distinct; a(n) = Sum_{k = 1..w} 2^(e_k mod m).

%C See A293390 for the corresponding m's.

%H Rémy Sigrist, <a href="/A356365/b356365.txt">Table of n, a(n) for n = 0..8192</a>

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

%F A000120(a(n)) = A000120(n).

%F a(n) = 1 iff n is a power of 2.

%F a(2^k - 1) = 2^k - 1 for any k >= 0.

%e The first terms, alongside their binary expansions and the corresponding m's, are:

%e n a(n) bin(n) bin(a(n)) m

%e --- ---- ------- --------- -

%e 0 0 0 0 0

%e 1 1 1 1 1

%e 2 1 10 1 1

%e 3 3 11 11 2

%e 4 1 100 1 1

%e 5 5 101 101 3

%e 6 3 110 11 2

%e 7 7 111 111 3

%e 8 1 1000 1 1

%e 9 3 1001 11 2

%e 10 3 1010 11 3

%e 11 11 1011 1011 4

%e 12 3 1100 11 2

%e 13 13 1101 1101 4

%e 14 7 1110 111 3

%e 15 15 1111 1111 4

%e 16 1 10000 1 1

%o (PARI) a(n) = { my (b=vector(hammingweight(n))); for (i=1, #b, n-=2^b[i]=valuation(n,2);); for (m=1, oo, if (#Set(b%m)==#b, b%=m; break;);); sum(i=1, #b, 2^b[i]); }

%Y Cf. A000120, A064895, A293390.

%K nonn,base

%O 0,4

%A _Rémy Sigrist_, Oct 16 2022