A239278 Smallest k > 1 such that n*(n+1)*...*(n+k-1) / (n+(n+1)+...+(n+k-1)) is an integer.

2, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 9, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3

Offset: 0

Author

Derek Orr, Mar 13 2014

Keywords

• nonn

Comments

a(n) = 7 for n == 8 (mod 15) (provided n != 53 (mod 105)).

a(n) = 5 for n == 2 (mod 3) (provided n != 8 (mod 15)).

a(n) = 9 for n == 53 (mod 105). - Jon E. Schoenfield, Mar 14 2014

a(n) = 3 for n == {0,1} (mod 3). - Zak Seidov, Mar 14 2014

Examples

			1*2/(1+2) = 2/3 is not an integer. 1*2*3/(1+2+3) = 1 is an integer. Thus a(1) = 3.
2*3/(2+3) = 6/5 is not an integer. 2*3*4/(2+3+4) = 24/9 is not an integer. 2*3*4*5/(2+3+4+5) = 120/14 is not an integer. 2*3*4*5*6/(2+3+4+5+6) = 720/20 = 36 is an integer. Thus a(2) = 5.
a(0) = 2 as 0*(0+(2-1)) / 0+(0+(2-1)) = 0/1 = 0 is an integer. - _Antti Karttunen_, Jan 18 2025
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..20000

Crossrefs

A284721 has the same start.

Programs

PARI

a(n) = {k = 2; while ( prod(i=0, k-1, n+i) % sum(i=0, k-1, n+i), k++); k;} \\ Michel Marcus, Mar 14 2014

PARI

A239278(n) = { my(m=n, s=n); for(k=2, oo, m *= (n+(k-1)); s += (n+(k-1)); if(!(m%s), return(k))); }; \\ Antti Karttunen, Jan 18 2025

Python

def A239278(n):
  (m, s, t) = (n, n, n+1)
  while 1:
    m *= t
    s += t
    if 0 == (m%s): return (1+t-n)
    else: t += 1
# Antti Karttunen, Jan 18 2025

Extensions

Term a(0)=2 prepended, original Python program replaced with one that works. - Antti Karttunen, Jan 18 2025