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%I #80 Dec 31 2023 06:23:36
%S 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,
%T 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,
%U 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
%N a(n) is the number of positive integers k such that k*(k+1) divides n.
%C The usual OEIS policy is not to include sequences like this where alternate terms are zero; this is an exception.
%C In other words, a(n) is the number of oblong numbers (A002378) dividing n. - _Bernard Schott_, Jul 29 2022
%H Ray Chandler, <a href="/A129308/b129308.txt">Table of n, a(n) for n = 1..10000</a>
%H P. Erdős and R. R. Hall, <a href="http://www.renyi.hu/~p_erdos/1978-26.pdf">On some unconventional problems on the divisors of integers</a>, J. Austral. Math. Soc., Ser. A, 25, 479-485 (1978).
%H MathOverflow, <a href="http://mathoverflow.net/questions/207512">On the number of consecutive divisors of an integer</a>.
%F a(2n-1) = 0; a(2n) = A007862(n). - _Ray Chandler_, Jun 24 2008
%F G.f.: Sum_{n>=1} x^(n*(n+1))/(1-x^(n*(n+1))). - _Joerg Arndt_, Jan 30 2011 [modified by _Ilya Gutkovskiy_, Apr 14 2021]
%F a(n) = A000005(n) - A137921(n), where A137921(n) is the number of maximal runs of successive divisors of n. - _Gus Wiseman_, Oct 15 2019
%F a(n) = Sum_{d|n} A005369(d). - _Ridouane Oudra_, Jan 22 2021
%F a(n) = A195155(n)-1. - _Antti Karttunen_, Feb 21 2023
%F From _Amiram Eldar_, Dec 31 2023: (Start)
%F a(n) = A088722(n) + A059841(n).
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1. (End)
%e 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.
%t 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 *)
%t Table[Count[Differences[Divisors[n]],1],{n,30}] (* _Gus Wiseman_, Oct 15 2019 *)
%o (PARI) a(n)=sumdiv(n, d, n%(d+1)==0); \\ _Michel Marcus_, Jan 06 2015
%Y Positions of 0's and 1's are A088725, whose characteristic function is A360128.
%Y First appearance of n is A287142(n), with sorted version A328450.
%Y The longest run of divisors of n has length A055874(n).
%Y One less than A195155.
%Y Cf. A000005, A002378, A003601, A007862, A027750, A033676, A060680, A060681, A072627, A088722, A181063, A199970, A328026, A328165, A328166.
%Y Cf. A005369, A059841, A287142, A344005.
%K nonn
%O 1,6
%A _Leroy Quet_, May 26 2007
%E More terms from _Stefan Steinerberger_, May 27 2007
%E Extended by _Ray Chandler_, Jun 24 2008
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