A212151 -id:A212151 - cp's OEIS frontend

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

Leroy Quet, Jun 29 2000

nonn

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

			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

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.

Cf. A000005, A000385, A072031, A212151.

Cf. A006218, A341062.

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

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

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

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

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

More terms from James Sellers, Jul 04 2000

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

A212240 Number of 2 X 2 matrices M with all terms in {1,...,n} and permanent(M) >= n.

Original entry on oeis.org

0, 1, 16, 80, 251, 612, 1269, 2354, 4021, 6449, 9844, 14427, 20458, 28203, 37972, 50073, 64876, 82725, 104046, 129222, 158741, 193024, 232607, 277956, 329675, 388248, 454353, 528508, 611435, 703712, 806121, 919242, 1043953, 1180865

Offset: 0

Clark Kimberling, May 07 2012

nonn

For a guide to related sequences, see A211795.

Cf. A211795, A212151.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w*x + y*z >= n, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]  (* A212240 *)
(* Peter J. C. Moses, Apr 13 2012 *)

from sympy import divisor_count
def A212240(n): return  n**4-sum((sum(divisor_count(i+1)*divisor_count(j-i) for i in range(j>>1))<<1)+(divisor_count(j+1>>1)**2 if j&1 else 0) for j in range(1,n-1)) # Chai Wah Wu, Jul 26 2024

a(n) + A212151(n) = n^4.

Offset changed to 0 by Georg Fischer, Feb 03 2022

A212241 Number of 2 X 2 matrices M with terms in {1,...,n} such that permanent(M) > n.

Original entry on oeis.org

0, 1, 15, 76, 243, 598, 1249, 2326, 3984, 6405, 9786, 14363, 20378, 28117, 37864, 49965, 64740, 82591, 103877, 129062, 158543, 192832, 232371, 277740, 329399, 388002, 454043, 528220, 611087, 703402, 805721, 918898, 1043520, 1180469

Offset: 0

Clark Kimberling, May 08 2012

nonn

a(n) + A212151(n+1) = n^4. For a guide to related sequences, see A211795.

Cf. A211795.

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w*x + y*z > n, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]  (* A212241 *)
(* Peter J. C. Moses, Apr 13 2012 *)

cp's OEIS Frontend

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

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A212240 Number of 2 X 2 matrices M with all terms in {1,...,n} and permanent(M) >= n.

Views

Author

Keywords

Comments

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Programs

Mathematica

Python

Formula

Extensions

A212241 Number of 2 X 2 matrices M with terms in {1,...,n} such that permanent(M) > n.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica