A152770 Sum of proper divisors minus the number of proper divisors of n: a(n) = sigma(n) - n - d(n) + 1.

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

Offset

1,6

Comments

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.

a(A008578(n)) = 0 for all n>=1. - Robert G. Wilson v, Dec 14 2008

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).

Formula

a(n) = A000203(n) - A000005(n) - n + 1 = A001065(n) - A000005(n) + 1 = A000203(n) - A062249(n) + 1 = A065608(n) - n + 1.

a(n) = A000203(n) - A032741(n) - n.

a(n) = A001065(n) - A032741(n).

a(n) = A158901(n) - n. - Juri-Stepan Gerasimov, Sep 12 2009

From Peter Bala Jan 22 2021: (Start)

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)

Maple

A152770 := proc(n)
        numtheory[sigma](n)-n-numtheory[tau](n)+1 ;
end proc: # R. J. Mathar, Sep 28 2011

Mathematica

f[n_] := DivisorSigma[1, n] - DivisorSigma[0, n] - n + 1; Array[f, 105] (* Robert G. Wilson v, Dec 14 2008 *)

Prog

(PARI) a(n)=sigma(n)-n-numdiv(n)+1 \\ Charles R Greathouse IV, Mar 09 2014

Crossrefs

Cf. A000005, A000040, A000203, A001065, A002808, A008578, A062249, A065608, A032741.

Sequence in context: A272192 A077150 A065453 * A098601 A113486 A108572

Adjacent sequences: A152767 A152768 A152769 * A152771 A152772 A152773

Keyword

nonn,easy

Author

Omar E. Pol, Dec 12 2008

Extensions

More terms from Omar E. Pol and Robert G. Wilson v, Dec 14 2008

Definition clarified and edited by Omar E. Pol, Dec 21 2008

Status

approved