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A092517
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Product of tau values for consecutive integers.
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13
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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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refs;
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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Ray Chandler, Table of n, a(n) for n = 1..10000
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FORMULA
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a(n) = A000005(n)*A000005(n+1) = A000005(n*(n+1)) = A000005(A002378(n)) = 2*A063123(n).
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MAPLE
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with(numtheory): seq(tau(n)*tau(n+1), n=1..73); # Zerinvary Lajos, Jan 22 2007
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A000005, A002378, A063123, A063440, A083539, A123000.
Sequence in context: A111973 A349787 A161655 * A128558 A090346 A267460
Adjacent sequences: A092514 A092515 A092516 * A092518 A092519 A092520
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KEYWORD
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nonn
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AUTHOR
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Reinhard Zumkeller, Apr 06 2004
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EXTENSIONS
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Extended by Ray Chandler, Jun 23 2008
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STATUS
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approved
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