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A129308
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a(n) is the number of positive integers k such that k*(k+1) divides n.
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29
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0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 3, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 3, 0, 1, 0, 2, 0, 3, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 5, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 4, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 4, 0, 1, 0, 1, 0, 4, 0, 1, 0, 1, 0, 3, 0, 1, 0, 2, 0, 2, 0, 1, 0
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OFFSET
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1,6
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COMMENTS
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The usual OEIS policy is not to include sequences like this where alternate terms are zero; this is an exception.
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LINKS
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FORMULA
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Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1. (End)
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EXAMPLE
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The divisors of 20 are 1,2,4,5,10,20. Of these there are two that are of the form k(k+1): 2 = 1*2 and 20 = 4*5. So a(2) = 2.
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MATHEMATICA
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a = {}; For[n = 1, n < 90, n++, k = 1; co = 0; While[k < Sqrt[n], If[IntegerQ[ n/(k*(k + 1))], co++ ]; k++ ]; AppendTo[a, co]]; a (* Stefan Steinerberger, May 27 2007 *)
Table[Count[Differences[Divisors[n]], 1], {n, 30}] (* Gus Wiseman, Oct 15 2019 *)
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PROG
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(PARI) a(n)=sumdiv(n, d, n%(d+1)==0); \\ Michel Marcus, Jan 06 2015
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CROSSREFS
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Positions of 0's and 1's are A088725, whose characteristic function is A360128.
The longest run of divisors of n has length A055874(n).
Cf. A000005, A002378, A003601, A007862, A027750, A033676, A060680, A060681, A072627, A088722, A181063, A199970, A328026, A328165, A328166.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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