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

Author

Omar E. Pol, Dec 12 2008

Keywords

• nonn
• easy

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).

Crossrefs

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

Programs

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 *)

PARI

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

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)

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