A055507 a(n) = Sum_{k=1..n} d(k)*d(n+1-k), where d(k) is number of positive divisors of k.

1, 4, 8, 14, 20, 28, 37, 44, 58, 64, 80, 86, 108, 108, 136, 134, 169, 160, 198, 192, 236, 216, 276, 246, 310, 288, 348, 310, 400, 344, 433, 396, 474, 408, 544, 450, 564, 512, 614, 522, 688, 560, 716, 638, 756, 636, 860, 676, 859, 772, 926, 758, 1016, 804, 1032

Offset: 1

Author

Leroy Quet, Jun 29 2000

Keywords

• nonn

Comments

a(n) is the number of ordered ways to express n+1 as a*b+c*d with 1 <= a,b,c,d <= n. - David W. Wilson, Jun 16 2003

tau(n) (A000005) convolved with itself, treating this result as a sequence whose offset is 2. - Graeme McRae, Jun 06 2006

Convolution of A341062 and nonzero terms of A006218. - Omar E. Pol, Feb 16 2021

Examples

			a(4) = d(1)*d(4) + d(2)*d(3) + d(3)*d(2) + d(4)*d(1) = 1*3 +2*2 +2*2 +3*1 = 14.
3 = 1*1+2*1 in 4 ways, so a(2)=4; 4 = 1*1+1*3 (4 ways) = 2*1+2*1 (4 ways), so a(3)=8; 5 = 4*1+1*1 (4 ways) = 2*2+1*1 (2 ways) + 3*1+2*1 (8 ways), so a(4) = 14. - _N. J. A. Sloane_, Jul 07 2012
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• George E. Andrews, Stacked lattice boxes, Ann. Comb. 3 (1999), 115-130. See D_{0,0}.
• A. E. Ingham, Some asymptotic formulae in the theory of numbers, Journal of the London Mathematical Society, Vol. s1-2, No. 3 (1927), pp. 202-208.
• Yoichi Motohashi, The binary additive divisor problem, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 27 no. 5 (1994), p. 529-572.
• E. C. Titchmarsh, Some problems in the analytic theory of numbers, The Quarterly Journal of Mathematics 1 (1942): 129-152.

Crossrefs

Cf. A000005, A000385, A072031, A212151.

Cf. A006218, A341062.

Programs

Maple

with(numtheory); A055507:=n->add(tau(j)*tau(n+1-j),j=1..n);

Mathematica

Table[Sum[DivisorSigma[0, k]*DivisorSigma[0, n + 1 - k], {k, 1, n}], {n, 1, 100}] (* Vaclav Kotesovec, Aug 08 2022 *)

PARI

a(n)=sum(k=1,n,numdiv(k)*numdiv(n+1-k)) \\ Charles R Greathouse IV, Oct 17 2012

Python

from sympy import divisor_count
def A055507(n): return  (sum(divisor_count(i+1)*divisor_count(n-i) for i in range(n>>1))<<1)+(divisor_count(n+1>>1)**2 if n&1 else 0) # Chai Wah Wu, Jul 26 2024

Formula

G.f.: Sum_{i >= 1, j >= 1} x^(i+j-1)/(1-x^i)/(1-x^j). - Vladeta Jovovic, Nov 11 2001

Working with an offset of 2, it appears that the o.g.f is equal to the Lambert series sum {n >= 2} A072031(n-1)*x^n/(1 - x^n). - Peter Bala, Dec 09 2014

a(n) = A212151(n+2) - A212151(n+1). - Ridouane Oudra, Sep 12 2020

Extensions

More terms from James Sellers, Jul 04 2000

Definition clarified by N. J. A. Sloane, Jul 07 2012