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

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, 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, 12, 11, 10, 9, 8, 7, 6, 5, 8, 7, 6, 11, 10, 9, 8, 7, 6, 5

Offset: 1

Author

Scott R. Shannon and N. J. A. Sloane, Feb 24 2020

Keywords

• nonn
• look

Comments

This is a multiplicative analog of A332542.

a(n) always exists because one can take k to be 2^m - 1 for m large.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• David A. Corneth, PARI program
• J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004.14000 [math.NT], April 2020.

Crossrefs

Cf. A061836 (k+1), A332559 (n+k+1), A332560 (the final product), A332561 (the quotient).

For records, see A333532 and A333533 (and A333537), which give the records in the essentially identical sequence A061836.

Additive version: A332542, A332543, A332544, A081123.

"Concatenate in base 10" version: A332580, A332584, A332585.

Programs

Maple

f:= proc(n) local k,p;
  p:= n;
  for k from 1 do
    p:= p*(n+k);
    if (p/(n+k+1))::integer then return k fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, Feb 25 2020

Mathematica

a[n_] := Module[{k, p = n}, For[k = 1, True, k++, p *= (n+k); If[Divisible[p, n+k+1], Return[k]]]];
Array[a, 100] (* Jean-François Alcover, Jun 04 2020, after Maple *)

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

PARI

\\ See Corneth link

Python

def a(n):
    k, p = 1, n*(n+1)
    while p%(n+k+1): k += 1; p *= (n+k)
    return k
print([a(n) for n in range(1, 85)]) # Michael S. Branicky, Jun 06 2021

Formula

a(n) = A061836(n) - 1 for n >= 1.

a(n + 1) >= a(n) - 1. a(n + 1) = a(n) - 1 mostly. - David A. Corneth, Apr 14 2020