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A383972

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

1, 3, 2, 3, 4, 3, 6, 7, 2, 4, 10, 3, 12, 7, 5, 7, 16, 3, 18, 4, 6, 11, 22, 8, 4, 12, 8, 7, 28, 15, 30, 15, 11, 16, 14, 3, 36, 19, 12, 15, 40, 20, 42, 11, 5, 23, 46, 8, 6, 4, 17, 12, 52, 8, 10, 7, 18, 28, 58, 15, 60, 31, 6, 15, 25, 11, 66, 16, 23, 20, 70, 8, 72, 36, 5, 19, 21, 12, 78

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OFFSET

1,2

COMMENTS

(m*(m + 1))^2 must be divisible by 4*n.

A011772: If m >= n, then n = 2^k. If m = n - 1, then n is a power of an odd prime. If m = n/2, then n is a prime of the form 4*k + 3.

this sequence: If m >= n, then n = 1 or n = 2. If m = n - 1, then n is an odd prime (Comment R. Israel). If m = n/2, then n = 2*p, p prime of the form 4*k + 3.

A383075: If m >= n, then n = 2^i or n = 3^j or n = 2^r * 3^s for some r, s. If m = n/2, then n = 2*p, p prime of the form 8*k + 3.

If n is an odd prime, a(n) = n-1. - Robert Israel, May 18 2025

LINKS

Michel Marcus, Table of n, a(n) for n = 1..10000

EXAMPLE

n = 2: smallest m such that (m*(m + 1))^2 is divisible by 4*2 is m = 3.

The first few numbers of the form (m*(m + 1))^2 / (4*n), m >= 1 are 1, 18, 3, 9, 20, 6, 63, ...

MAPLE

f:= proc(n) local x, R;

R:= map(t -> rhs(op(t)), [msolve((x*(x+1))^2=0, 4*n)]);

min(subs(0=4*n, R))

end proc: