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A152770
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Sum of proper divisors minus the number of proper divisors of n: a(n) = sigma(n) - n - d(n) + 1.
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22
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0, 0, 0, 1, 0, 3, 0, 4, 2, 5, 0, 11, 0, 7, 6, 11, 0, 16, 0, 17, 8, 11, 0, 29, 4, 13, 10, 23, 0, 35, 0, 26, 12, 17, 10, 47, 0, 19, 14, 43, 0, 47, 0, 35, 28, 23, 0, 67, 6, 38, 18, 41, 0, 59, 14, 57, 20, 29, 0, 97, 0, 31, 36, 57, 16, 71, 0, 53, 24, 67, 0, 112, 0, 37, 44, 59, 16, 83, 0, 97
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OFFSET
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1,6
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COMMENTS
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Sum of divisors of n, minus the number of divisors of n, minus n, plus 1.
Also, sum of proper divisors of n, minus the number of divisors of n, plus 1.
Note that if a(n)>0 then n is a composite number (A002808), otherwise, n is a noncomposite number (A008578) also called prime number at the beginning of the 20th century.
Also, sum of divisors of n, minus the number of proper divisors of n, minus n.
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LINKS
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FORMULA
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G.f.: A(q) = Sum_{n >= 2} (n-1)*q^(2*n)/(1 - q^n) = Sum_{n >= 2} q^(2*n)/(1 - q^n)^2. Cf. A001065.
Faster converging series: A(q) = Sum_{n >= 1} q^(n*(n+1))*((n-1)*q^(3*n+2) - n*q^(2*n+1) + (2-n)*q^(n+1) + n - 1)/((1 - q^n)*(1 - q^(n+1))^2) - apply the operator t*d/dt to equation 1 in Arndt, then set t = q^2 and x = q. (End)
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MAPLE
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numtheory[sigma](n)-n-numtheory[tau](n)+1 ;
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MATHEMATICA
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f[n_] := DivisorSigma[1, n] - DivisorSigma[0, n] - n + 1; Array[f, 105] (* Robert G. Wilson v, Dec 14 2008 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Definition clarified and edited by Omar E. Pol, Dec 21 2008
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STATUS
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approved
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