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

Author

Reinhard Zumkeller, Apr 06 2004

Keywords

• nonn

Comments

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

Links

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

Crossrefs

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

Programs

Magma

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

Maple

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

Mathematica

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 *)

PARI

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

Python

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

Formula

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

Extensions

Extended by Ray Chandler, Jun 23 2008