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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A383972 Smallest number m such that (m*(m + 1)/2)^2 is divisible by n.

Original entry on oeis.org

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
Offset: 1

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Author

Ctibor O. Zizka, May 16 2025

Keywords

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

Examples

			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, ...

Crossrefs

Cf. A000537, A002145, A007520, A011772, A383075.

Programs

  • 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:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, May 18 2025
  • Mathematica
    a[n_] := Module[{m = 1}, While[PowerMod[m*(m + 1)/2, 2, n] > 0, m++]; m]; Array[a, 100] (* Amiram Eldar, May 17 2025 *)
  • PARI
    a(n) = my(m=1); while (Mod(m*(m+1)/2, n)^2, m++); m; \\ Michel Marcus, May 16 2025
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