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A246601
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Sum of divisors d of n with property that the binary representation of d can be obtained from the binary representation of n by changing any number of 1's to 0's.
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6
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1, 2, 4, 4, 6, 8, 8, 8, 10, 12, 12, 16, 14, 16, 24, 16, 18, 20, 20, 24, 22, 24, 24, 32, 26, 28, 40, 32, 30, 48, 32, 32, 34, 36, 36, 40, 38, 40, 43, 48, 42, 44, 44, 48, 60, 48, 48, 64, 50, 52, 72, 56, 54, 80, 61, 64, 58, 60, 60, 96, 62, 64, 104, 64, 66, 68, 68, 72, 70, 72
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OFFSET
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1,2
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COMMENTS
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Equivalently, the sum of the divisors d of n such that the bitwise OR of d and n is equal to n. - Chai Wah Wu, Sep 06 2014
Equivalently, the sum of the divisors d of n such that the bitwise AND of n and d is equal to d. - Amiram Eldar, Dec 15 2022
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LINKS
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FORMULA
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a(2^i) = 2^i.
a(odd prime p) = p+1.
a(2*n) = 2*a(n), and therefore a(m*2^k) = 2^k*a(m) for m odd and k>=0.
a(2^n-1) = sigma(2^n-1) = A075708(n). (End)
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EXAMPLE
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12 = 1100_2; only the divisors 4 = 0100_2 and 12 = 1100_2 satisfy the condition, so(12) = 4+12 = 16.
15 = 1111_2; all divisors 1,3,5,15 satisfy the condition, so a(15)=24.
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MAPLE
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with(numtheory);
sd:=proc(n) local a, d, s, t, i, sw;
s:=convert(n, base, 2);
a:=0;
for d in divisors(n) do
sw:=-1;
t:=convert(d, base, 2);
for i from 1 to nops(t) do if t[i]>s[i] then sw:=1; fi; od:
if sw<0 then a:=a+d; fi;
od;
a;
end;
[seq(sd(n), n=1..100)];
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MATHEMATICA
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a[n_] := DivisorSum[n, #*Boole[BitOr[#, n] == n] &]; Array[a, 100] (* Jean-François Alcover, Dec 02 2015, adapted from PARI *)
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PROG
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(Python)
from sympy import divisors
....return sum(d for d in divisors(n) if n|d == n)
(PARI) a(n) = sumdiv(n, d, d*(bitor(n, d)==n)); \\ Michel Marcus, Sep 07 2014
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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