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

Author

Darío Clavijo, Sep 15 2023

Keywords

• nonn

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.

Examples

			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.
		

Links

• Project Euler, Problem 261: Pivotal Square Sums.

Crossrefs

Cf. A000290, A002378, A046092, A233035.

Programs

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

Python

from sympy import isprime
from sympy.ntheory.primetest import is_square
from sympy.core.intfunc import isqrt
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 isprime(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+1, m * m, m << 1
        S = k2 - k * A046092(m)
        while S > 0:
            S -= m2 + (h * m_2)
            h += 1
            if S == 0: return True
print([k for k in range(1, 3256) if isok(k)])

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.