cp's OEIS Frontend

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

%I A152770 #28 Jan 24 2021 10:17:27
%S A152770 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,
%T A152770 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,
%U A152770 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
%N A152770 Sum of proper divisors minus the number of proper divisors of n: a(n) = sigma(n) - n - d(n) + 1.
%C A152770 Sum of divisors of n, minus the number of divisors of n, minus n, plus 1.
%C A152770 Also, sum of proper divisors of n, minus the number of divisors of n, plus 1.
%C A152770 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.
%C A152770 Also, sum of divisors of n, minus the number of proper divisors of n, minus n.
%C A152770 a(A008578(n)) = 0 for all n>=1. - _Robert G. Wilson v_, Dec 14 2008
%H A152770 Vincenzo Librandi, <a href="/A152770/b152770.txt">Table of n, a(n) for n = 1..1000</a>
%H A152770 Joerg Arndt, <a href="http://arxiv.org/abs/1202.6525">On computing the generalized Lambert series</a>, arXiv:1202.6525v3 [math.CA], (2012).
%F A152770 a(n) = A000203(n) - A000005(n) - n + 1 = A001065(n) - A000005(n) + 1 = A000203(n) - A062249(n) + 1 = A065608(n) - n + 1.
%F A152770 a(n) = A000203(n) - A032741(n) - n.
%F A152770 a(n) = A001065(n) - A032741(n).
%F A152770 a(n) = A158901(n) - n. - _Juri-Stepan Gerasimov_, Sep 12 2009
%F A152770 From _Peter Bala_ Jan 22 2021: (Start)
%F A152770 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.
%F A152770 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)
%p A152770 A152770 := proc(n)
%p A152770         numtheory[sigma](n)-n-numtheory[tau](n)+1 ;
%p A152770 end proc: # _R. J. Mathar_, Sep 28 2011
%t A152770 f[n_] := DivisorSigma[1, n] - DivisorSigma[0, n] - n + 1; Array[f, 105] (* _Robert G. Wilson v_, Dec 14 2008 *)
%o A152770 (PARI) a(n)=sigma(n)-n-numdiv(n)+1 \\ _Charles R Greathouse IV_, Mar 09 2014
%Y A152770 Cf. A000005, A000040, A000203, A001065, A002808, A008578, A062249, A065608, A032741.
%K A152770 nonn,easy
%O A152770 1,6
%A A152770 _Omar E. Pol_, Dec 12 2008
%E A152770 More terms from _Omar E. Pol_ and _Robert G. Wilson v_, Dec 14 2008
%E A152770 Definition clarified and edited by _Omar E. Pol_, Dec 21 2008