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A351112
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Number of balanced numbers dividing n.
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4
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1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 5, 1, 3, 3, 2, 1, 4, 1, 2, 2, 2, 1, 5, 1, 2, 2, 3, 1, 6, 1, 2, 2, 2, 2, 5, 1, 2, 2, 2, 1, 6, 1, 2, 3, 2, 1, 5, 1, 2, 2, 2, 1, 4, 1, 4, 2, 2, 1, 7, 1, 2, 2, 2, 1, 4, 1, 2, 2, 5, 1, 5, 1, 2, 3, 2, 1, 5, 1, 2, 2, 2, 1, 7, 1, 2, 2, 2, 1, 6, 1, 2
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OFFSET
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1,2
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COMMENTS
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A balanced number k is a number such that phi(k) | sigma(k).
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LINKS
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FORMULA
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a(n) = Sum_{d|n, phi(d)|sigma(d)} 1.
a(n) = tau(n) - Sum_{d|n} sign(sigma(d) mod phi(d)).
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EXAMPLE
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a(4) = 2; the balanced divisors of 4 are 1 and 2.
a(5) = 1; 1 is the only balanced divisor of 5.
a(6) = 4; the balanced divisors of 6 are 1,2,3,6.
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MAPLE
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f:= proc(n) uses numtheory;
nops(select(t -> sigma(t) mod phi(t) = 0, divisors(n)))
end proc:
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MATHEMATICA
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a[n_] := DivisorSum[n, 1 &, Divisible[DivisorSigma[1, #], EulerPhi[#]] &]; Array[a, 100] (* Amiram Eldar, Feb 01 2022 *)
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PROG
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(PARI) a(n) = sumdiv(n, d, if (!(sigma(d) % eulerphi(d)), 1)); \\ Michel Marcus, Feb 01 2022
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CROSSREFS
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Cf. A351113 (sum of the balanced numbers dividing n).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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