A383075 - OEIS

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

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OFFSET

1,2

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

LINKS

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

David A. Corneth, PARI program

EXAMPLE

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

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 *)

PROG

(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

CROSSREFS

Cf. A000330, A007520, A011772.

Sequence in context: A294896 A021292 A230159 * A105828 A209873 A130880