A383075 Smallest number m such that m*(m + 1)*(2*m + 1)/6 is divisible by n.

1, 3, 4, 7, 2, 4, 3, 15, 13, 4, 5, 8, 6, 3, 4, 31, 8, 27, 9, 7, 13, 11, 11, 31, 12, 12, 40, 7, 14, 4, 15, 63, 22, 8, 7, 40, 18, 19, 13, 15, 20, 27, 21, 16, 27, 11, 23, 31, 24, 12, 8, 32, 26, 40, 5, 31, 9, 28, 29, 40, 30, 15, 13, 127, 12, 27, 33, 8, 22, 7, 35

Offset: 1

Author

Ctibor O. Zizka, Apr 15 2025

Keywords

• nonn

Comments

m*(m + 1)*(2*m + 1)/6 is divisible by n if and only if m*(m + 1)*(2*m + 1) is divisible by 6*n. - David A. Corneth, Apr 21 2025

If n > 3 is a prime of the form (8*k + 3), then a(2*n) = n. - Ctibor O. Zizka, May 21 2025

Examples

			n = 2: smallest m such that m*(m + 1)*(2*m + 1) is divisible by 2*6 is m = 3.
a(4) = 7. The first few numbers of the form m*(m + 1)*(2*m + 1)/6, m >= 1 are 1, 5, 14, 30, 55, 91, 140,... The first 6 are not divisible by 4 but the seventh is. - _David A. Corneth_, Apr 21 2025
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000
• David A. Corneth, PARI program

Crossrefs

Cf. A000330, A007520, A011772.

Programs

Mathematica

a[n_]:=Module[{m=1},While[!Divisible[m(m+1)(2m+1)/6,n], m++]; m]; Array[a,71] (* Stefano Spezia, Apr 15 2025 *)

PARI

a(n) = my(m=1); while (m*(m+1)*(2*m+1)/6 % n, m++); m; \\ Michel Marcus, Apr 20 2025

PARI

\\ See Corneth link

Python

from itertools import count
from sympy import integer_nthroot
def A383075(n): return next(m for m in count(integer_nthroot(3*n,3)[0]) if not m*(m+1)*((m<<1)+1)%(6*n)) # Chai Wah Wu, Apr 21 2025