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A332558 a(n) is the smallest k such that n*(n+1)*(n+2)*...*(n+k) is divisible by n+k+1.

%I A332558 #39 Jun 06 2021 02:53:54
%S A332558 4,3,2,3,4,5,4,3,5,4,6,5,6,5,4,7,6,5,4,3,6,7,6,5,4,8,7,6,6,5,8,7,6,5,
%T A332558 4,8,7,6,5,7,6,5,10,9,8,9,8,7,6,9,8,7,6,5,4,6,12,11,10,9,8,7,6,7,6,5,
%U A332558 12,11,10,9,8,7,6,5,8,7,6,11,10,9,8,7,6,5
%N A332558 a(n) is the smallest k such that n*(n+1)*(n+2)*...*(n+k) is divisible by n+k+1.
%C A332558 This is a multiplicative analog of A332542.
%C A332558 a(n) always exists because one can take k to be 2^m - 1 for m large.
%H A332558 Robert Israel, <a href="/A332558/b332558.txt">Table of n, a(n) for n = 1..10000</a>
%H A332558 David A. Corneth, <a href="/A332558/a332558.gp.txt">PARI program</a>
%H A332558 J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, <a href="http://arxiv.org/abs/2004.14000">Three Cousins of Recaman's Sequence</a>, arXiv:2004.14000 [math.NT], April 2020.
%F A332558 a(n) = A061836(n) - 1 for n >= 1.
%F A332558 a(n + 1) >= a(n) - 1. a(n + 1) = a(n) - 1 mostly. - _David A. Corneth_, Apr 14 2020
%p A332558 f:= proc(n) local k,p;
%p A332558   p:= n;
%p A332558   for k from 1 do
%p A332558     p:= p*(n+k);
%p A332558     if (p/(n+k+1))::integer then return k fi
%p A332558   od
%p A332558 end proc:
%p A332558 map(f, [$1..100]); # _Robert Israel_, Feb 25 2020
%t A332558 a[n_] := Module[{k, p = n}, For[k = 1, True, k++, p *= (n+k); If[Divisible[p, n+k+1], Return[k]]]];
%t A332558 Array[a, 100] (* _Jean-François Alcover_, Jun 04 2020, after Maple *)
%o A332558 (PARI) a(n) = {my(r=n*(n+1)); for(k=2, oo, r=r*(n+k); if(r%(n+k+1)==0, return(k))); } \\ _Jinyuan Wang_, Feb 25 2020
%o A332558 (PARI) \\ See Corneth link
%o A332558 (Python)
%o A332558 def a(n):
%o A332558     k, p = 1, n*(n+1)
%o A332558     while p%(n+k+1): k += 1; p *= (n+k)
%o A332558     return k
%o A332558 print([a(n) for n in range(1, 85)]) # _Michael S. Branicky_, Jun 06 2021
%Y A332558 Cf. A061836 (k+1), A332559 (n+k+1), A332560 (the final product), A332561 (the quotient).
%Y A332558 For records, see A333532 and A333533 (and A333537), which give the records in the essentially identical sequence A061836.
%Y A332558 Additive version: A332542, A332543, A332544, A081123.
%Y A332558 "Concatenate in base 10" version: A332580, A332584, A332585.
%K A332558 nonn,look
%O A332558 1,1
%A A332558 _Scott R. Shannon_ and _N. J. A. Sloane_, Feb 24 2020