A063123 -id:A063123 - cp's OEIS frontend

A092517 Product of tau values for consecutive integers.

2, 4, 6, 6, 8, 8, 8, 12, 12, 8, 12, 12, 8, 16, 20, 10, 12, 12, 12, 24, 16, 8, 16, 24, 12, 16, 24, 12, 16, 16, 12, 24, 16, 16, 36, 18, 8, 16, 32, 16, 16, 16, 12, 36, 24, 8, 20, 30, 18, 24, 24, 12, 16, 32, 32, 32, 16, 8, 24, 24, 8, 24, 42, 28, 32, 16, 12, 24, 32, 16, 24, 24, 8, 24, 36

Offset: 1

Reinhard Zumkeller, Apr 06 2004

nonn

Number of divisors of the n-th oblong number. - Ray Chandler, Jun 23 2008
Number of positive solutions (x,y) for which n/x + (n+1)/y = 1. - Michel Lagneau, Jan 16 2014
Number of positive solutions for which 1/p + 1/q + 1/(p*q) = 1/n; set p=x and q=y-1 in the solutions (x,y) in the comment above. - Mo Li, Apr 27 2021
a(n) is the maximum number of b > 0, which allows us to write (n+1)^2 as a sum of n+1 parts. Each part is of the form b^c and c is an integer >= 0 independent for each part. For n = 2 this is 3^2 = 2^2 + 2^2 + 2^0 = 3^1 + 3^1 + 3^1 = 4^1 + 4^1 + 4^0 = 7^1 + 7^0 + 7^0, b = 2;3;4;7 and a(2) = 4. It is conjectured that for all n the number of possible b reaches a(n). - Thomas Scheuerle, Jan 12 2022

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A000005, A002378, A063123, A063440, A083539, A123000.

[ NumberOfDivisors(n^2+n) : n in [1..100]]; // Vincenzo Librandi, Apr 03 2011

with(numtheory): seq(tau(n)*tau(n+1),n=1..73); # Zerinvary Lajos, Jan 22 2007

Table[DivisorSigma[0,n^2+n],{n,100}] (* Giorgos Kalogeropoulos, Apr 28 2021 *)
Times@@#&/@Partition[DivisorSigma[0,Range[80]],2,1] (* Harvey P. Dale, Apr 21 2022 *)

a(n) = numdiv(n^2+n); \\ Michel Marcus, Jan 11 2020

from sympy import divisor_count
def A092517(n): return divisor_count(n)*divisor_count(n+1) # Chai Wah Wu, Jan 06 2022

a(n) = A000005(n)*A000005(n+1) = A000005(n*(n+1)) = A000005(A002378(n)) = 2*A063123(n).

Extended by Ray Chandler, Jun 23 2008

A330320 a(n) = Sum_{i=1..n} tau(i)*tau(i+1), where tau(n) = A000005(n) is the number of divisors of n.

Original entry on oeis.org

2, 6, 12, 18, 26, 34, 42, 54, 66, 74, 86, 98, 106, 122, 142, 152, 164, 176, 188, 212, 228, 236, 252, 276, 288, 304, 328, 340, 356, 372, 384, 408, 424, 440, 476, 494, 502, 518, 550, 566, 582, 598, 610, 646, 670, 678, 698, 728, 746, 770, 794, 806, 822, 854, 886, 918, 934, 942, 966, 990, 998, 1022, 1064, 1092, 1124, 1140

Offset: 1

N. J. A. Sloane, Dec 11 2019

nonn

For background references see A330570.

József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 61.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
A. E. Ingham, Some asymptotic formulae in the theory of numbers, Journal of the London Mathematical Society, Vol. 1, No. 3 (1927), pp. 202-208.
Nikolay V. Kuznetsov, Convolution of the Fourier coefficients of the Eisenstein-Maass series, Journal of Soviet Mathematics, Vol. 29, No. 2 (1985), pp. 1131-1159.

Cf. A000005, A059956, A063123.

Partial sums of A092517.

Accumulate[a[n_]:=DivisorSum[n+1, DivisorSigma[0, n]&]; Array[a, 66]] (* Vincenzo Librandi, Jan 10 2020 *)
Accumulate[Times@@@Partition[DivisorSigma[0,Range[70]],2,1]] (* Harvey P. Dale, Nov 02 2023 *)

a(n) = sum(i=1, n, numdiv(i*(i+1))); \\ Michel Marcus, Jan 11 2020

a(n) ~ (1/zeta(2)) * n * log(n)^2. - Amiram Eldar, Mar 05 2020

A330321 a(n) = Sum_{i=1..n} tau(i)*tau(i+1)/2, where tau(n) = A000005(n) is the number of divisors of n.

Original entry on oeis.org

1, 3, 6, 9, 13, 17, 21, 27, 33, 37, 43, 49, 53, 61, 71, 76, 82, 88, 94, 106, 114, 118, 126, 138, 144, 152, 164, 170, 178, 186, 192, 204, 212, 220, 238, 247, 251, 259, 275, 283, 291, 299, 305, 323, 335, 339, 349, 364, 373, 385, 397, 403, 411, 427, 443, 459, 467, 471, 483, 495, 499, 511, 532, 546, 562, 570, 576, 588

Offset: 1

N. J. A. Sloane, Dec 11 2019

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A000005, A092517.

Partial sums of A063123.

Accumulate[a[n_]:=DivisorSum[n, DivisorSigma[0, n+1] / 2 &]; Array[a, 68]] (* Vincenzo Librandi, Jan 11 2020 *)

lista(nmax) = {my(d1 = 1, d2, s = 0); for(k = 2, nmax, d2 = numdiv(k); s += (d1 * d2 / 2); print1(s, ", "); d1 = d2);} \\ Amiram Eldar, Apr 19 2024

From Amiram Eldar, Apr 19 2024: (Start)
a(n) = A330320(n)/2.
a(n) ~ (3/Pi^2) * n * log(n)^2. (End)

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A092517 Product of tau values for consecutive integers.

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A330320 a(n) = Sum_{i=1..n} tau(i)*tau(i+1), where tau(n) = A000005(n) is the number of divisors of n.

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A330321 a(n) = Sum_{i=1..n} tau(i)*tau(i+1)/2, where tau(n) = A000005(n) is the number of divisors of n.

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