A158901 -id:A158901 - 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.

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

Omar E. Pol, Dec 12 2008

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

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).

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

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

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

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

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)

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

A158902 Triangle read by rows: the matrix product A051731 * A158821.

Original entry on oeis.org

1, 2, 1, 3, 0, 1, 5, 1, 0, 1, 5, 0, 0, 0, 1, 9, 1, 1, 0, 0, 1, 7, 0, 0, 0, 0, 0, 1, 12, 1, 0, 1, 0, 0, 0, 1, 11, 0, 1, 0, 0, 0, 0, 0, 1, 15, 1, 0, 0, 1, 0, 0, 0, 0, 1, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 23, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson and Mats Granvik, Mar 29 2009

			First few rows of the triangle =
1;
2, 1;
3, 0, 1;
5, 1, 0, 1;
5, 0, 0, 0, 1;
9, 1, 1, 0, 0, 1;
7, 0, 0, 0, 0, 0, 1;
12, 1, 0, 1, 0, 0, 0, 1
11, 0, 1, 0, 0, 0, 0, 0, 1;
15, 1, 0, 0, 1, 0, 0, 0, 0, 1;
11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
23, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1;
13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
21, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1;
...

Cf. A158821, A051731, A158901, A000203 (row sums).

A158902 := proc(n,k)
    add( A051731(n,j)*A158821(j-1,k-1),j=k..n) ;
end proc:
seq(seq(A158902(n,k),k=1..n),n=1..12) ; # R. J. Mathar, Jan 08 2015

m = 12; (* number of rows *)
T1[n_, k_] := Boole[Mod[n, k] == 0];
T2[n_, k_] := Which[n == k, 1, k == 1, n-1, True, 0];
T = Array[T1, {m, m}].Array[T2, {m, m}];
Table[T[[n, k]], {n, m}, {k, n}] // Flatten (* Jean-François Alcover, Feb 01 2023 *)

T(n,n) = 1.

T(n,1) = A158901(n).

Wrong A-number in definition corrected by Robert Israel, Jan 08 2015

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.

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