login
A092517
Product of tau values for consecutive integers.
16
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
1,1
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
FORMULA
a(n) = A000005(n)*A000005(n+1) = A000005(n*(n+1)) = A000005(A002378(n)) = 2*A063123(n).
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 *)
PROG
(Magma) [ NumberOfDivisors(n^2+n) : n in [1..100]]; // Vincenzo Librandi, Apr 03 2011
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Apr 06 2004
EXTENSIONS
Extended by Ray Chandler, Jun 23 2008
STATUS
approved