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a(n) is the sum of the divisors of n whose binary expansions appear as substrings in the binary expansion of n.
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%I #17 Jul 16 2022 07:18:04

%S 1,3,4,7,6,12,8,15,10,18,12,28,14,24,19,31,18,30,20,42,22,36,24,60,26,

%T 42,31,56,30,57,32,63,34,54,36,70,38,60,43,90,42,66,44,84,54,72,48,

%U 124,50,78,55,98,54,93,72,120,61,90,60,133,62,96,74,127,66

%N a(n) is the sum of the divisors of n whose binary expansions appear as substrings in the binary expansion of n.

%H Rémy Sigrist, <a href="/A355633/b355633.txt">Table of n, a(n) for n = 1..10000</a>

%H Rémy Sigrist, <a href="/A355633/a355633.png">Colored scatterplot of the first 100000 terms</a> (the color is function of the 2-adic valuation of n)

%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</a>

%F a(n) <= A000203(n).

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

%e For n = 84:

%e - the binary expansion of 84 is "1010100",

%e - we have the following divisors:

%e d bin(d) in bin(84)?

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

%e 1 1 Yes

%e 2 10 Yes

%e 3 11 No

%e 4 100 Yes

%e 6 110 No

%e 7 111 No

%e 12 1100 No

%e 14 1110 No

%e 21 10101 Yes

%e 28 11100 No

%e 42 101010 Yes

%e 84 1010100 Yes

%e - so a(84) = 1 + 2 + 4 + 21 + 42 + 84 = 154.

%t a[n_] := DivisorSum[n, # &, StringContainsQ @@ IntegerString[{n, #}, 2] &]; Array[a, 100] (* _Amiram Eldar_, Jul 16 2022 *)

%o (PARI) a(n, base=2) = { my (d=digits(n, base), s=setbinop((i, j) -> fromdigits(d[i..j], base), [1..#d]), v=0); for (k=1, #s, if (s[k] && n%s[k]==0, v+=s[k])); return (v) }

%o (Python)

%o from sympy import divisors

%o def a(n):

%o s = bin(n)[2:]

%o return sum(d for d in divisors(n, generator=True) if bin(d)[2:] in s)

%o print([a(n) for n in range(1, 66)]) # _Michael S. Branicky_, Jul 11 2022

%Y Cf. A000203, A027750, A093640, A355620 (decimal analog), A355634.

%K nonn,base

%O 1,2

%A _Rémy Sigrist_, Jul 11 2022