A365686 Numbers k such that there exists a pair of integers (m,h) where 1 <= m < floor(sqrt(k)/2) <= h that satisfy Sum_{j=0..m} (k-j)^2 = Sum_{i=1..m} (h+i)^2.

4, 12, 21, 24, 40, 60, 84, 110, 112, 120, 144, 180, 220, 264, 312, 315, 364, 420, 480, 544, 612, 684, 697, 760, 820, 840, 924, 1012, 1080, 1104, 1200, 1265, 1300, 1404, 1512, 1624, 1740, 1860, 1984, 2106, 2112, 2244, 2380, 2520, 2664, 2812, 2964, 3120, 3255

Offset

1,1

Comments

The sums are of m+1 consecutive squares ending at k^2, and of m consecutive squares starting somewhere at or beyond (k+1)^2.

If k is a term and h = k then k is in A046092.

All terms are composite numbers.

Also k is a term if there exists a pair of integers (m, h) such that 1 <= m < floor(sqrt(k)/2) <= h and that satisfy k*(m+1)*(k-m)-m*h*(h+m+1)=0.

Links

Table of n, a(n) for n=1..49.

Project Euler, Problem 261: Pivotal Square Sums.

Formula

k if Sum_{j=0..m} (k-j)^2 = Sum_{i=1..m} (h+i)^2 where 1 <= m < floor(sqrt(k)/2) <= h.

Example

k=24 is a term because 21^2 + 22^2 + 23^2 + 24^2 = 25^2 + 26^2 + 27^2 with m=3 and h=24.
k=110 is a term because 108^2 + 109^2 + 110^2 = 133^2 + 134^2, with m=2 and h=132.

Prog

(Python)
from gmpy2 import *
A002378 = lambda n: n * (n + 1)
A046092 = lambda n: A002378(n) << 1
isA046092 = lambda n: (n & 1 == 0) and is_square((n << 1) + 1)
def isok(k):
  if is_prime(k): return False
  if isA046092(k): return True
  k2 = k * k
  for m in range(1, (isqrt(k) >> 1) + 1):
    h, m2, m_2 = k, m * m, m << 1
    S = k2 - A046092(m) * k
    while(S > 0):
      h += 1
      S -= m2 + (h * m_2)
    if S == 0: return True
  return False
print([k for k in range(1, 3256) if isok(k)])
(PARI) isok(k) = for (i=1, k-1, my(s1 = sum(j=k-i, k, j^2)); for (m=k+1, oo, my(s2 = sum(j=0, i-1, (m+j)^2)); if (s2 == s1, return(1)); if (s2 > s1, break); ); ); \\ Michel Marcus, Sep 27 2023

Crossrefs

Cf. A000290, A002378, A046092, A233035.

Sequence in context: A087080 A134253 A115106 * A270795 A191260 A047965

Adjacent sequences: A365683 A365684 A365685 * A365687 A365688 A365689

Keyword

nonn

Author

Darío Clavijo, Sep 15 2023

Status

approved