A393157 - OEIS

%I #27 Feb 19 2026 12:46:04

%S 1,3,4,7,6,4,3,15,13,12,5,8,6,3,18,31,2,27,9,24,13,11,11,31,6,12,40,7,

%T 14,31,15,63,22,8,6,40,14,19,13,31,18,27,8,16,81,11,10,31,3,124,8,32,

%U 26,40,43,31,9,28,4,31,30,15,13,127,6,27,19,8,22,24,35

%N Smallest positive number m such that A000538(m) is divisible by n.

%H Paolo P. Lava, <a href="/A393157/b393157.txt">Table of n, a(n) for n = 1..10000</a>

%F a(2^j) = 2^(j+1) - 1.

%F a(p^j) = (p^(j+1) - 1)/2 for primes 3, 11, ...

%F a(p^j) = (p^j - 1)/2 for primes 7, 13, 19, 23, 29, ...

%p F:= proc(n) local x; min(map(t -> rhs(op(t)), subs({x=0}=NULL, [msolve(x*(x+1)*(2*x+1)*(3*x^2+3*x-1)/30,n)]))) end proc:

%p F(1):= 1:

%p map(F, [$1..120]); # _Robert Israel_, Feb 11 2026

%t Table[s = k = 1; While[! Divisible[s, n], k++; s += k^4]; k, {n, 120}] (* _Michael De Vlieger_, Feb 11 2026 *)

%o (PARI) a(n) = my(k=1); while (k*(1+k)*(1+2*k)*(-1+3*k+3*k^2)/30 % n, k++); k; \\ _Michel Marcus_, Feb 10 2026

%o (Python)

%o from itertools import count

%o def A393157(n): return next(m for m in count(1) if not m*(m**2*(m*(6*m+15)+10)-1)//30 % n) # _Chai Wah Wu_, Feb 17 2026

%Y Cf. A000538, A011772, A383075, A383972.

%K nonn,easy

%O 1,2

%A _Paolo P. Lava_, Feb 03 2026