

A092517


Product of tau values for consecutive integers.


13



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
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OFFSET

1,1


COMMENTS

Number of divisors of the nth 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=y1 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


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

Cf. A000005, A002378, A063123, A063440, A083539, A123000.
Sequence in context: A111973 A349787 A161655 * A128558 A090346 A267460
Adjacent sequences: A092514 A092515 A092516 * A092518 A092519 A092520


KEYWORD

nonn


AUTHOR

Reinhard Zumkeller, Apr 06 2004


EXTENSIONS

Extended by Ray Chandler, Jun 23 2008


STATUS

approved



