A356365 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).
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, 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, 57, 27, 59, 15, 61, 31, 63, 1, 5, 3, 7, 5, 7, 7, 71, 3, 13, 14, 15
Offset: 0
Examples
The first terms, alongside their binary expansions and the corresponding m's, are:
n a(n) bin(n) bin(a(n)) m
--- ---- ------- --------- -
0 0 0 0 0
1 1 1 1 1
2 1 10 1 1
3 3 11 11 2
4 1 100 1 1
5 5 101 101 3
6 3 110 11 2
7 7 111 111 3
8 1 1000 1 1
9 3 1001 11 2
10 3 1010 11 3
11 11 1011 1011 4
12 3 1100 11 2
13 13 1101 1101 4
14 7 1110 111 3
15 15 1111 1111 4
16 1 10000 1 1
Links
Programs
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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]); }
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