A346163 Numbers k such that there exist equal sums of k and 2k consecutive positive squares.

1, 17, 23, 25, 49, 55, 71, 73, 79, 89, 95, 103, 113, 127, 143, 161, 167, 175, 185, 191, 193, 199, 215, 217, 233, 239, 241, 265, 271, 287, 289, 305, 361, 377, 391, 409, 415, 431, 433, 457, 473, 481, 505, 511, 521, 535, 545, 553, 569, 593, 599, 617, 631, 647

Offset

1,2

Comments

a(n) is congruent to 1 or 5 (mod 6).

a(n) is not congruent to 3, 4 or 5 (mod 8) or to 7, 11, 16 or 20 (mod 27), see Alder and Alfred.

k is in the sequence if the quadratic Diophantine equation 6*(k*x^2 - 2*k*y^2 + k*(k-1)*x + 2*k*(1-2*k)*y) - 14*k^3 + 9*k^2 - k = 0 has solutions x, y in the positive integers.

Links

Table of n, a(n) for n=1..54.

H. L. Alder and U. Alfred, n and n+k Consecutive Integers with Equal Sums of Squares, The American Mathematical Monthly, 71:7 (1964), 749-754.

Index entries for sequences related to sums of squares

Example

a(1): 5^2 = 3^2 + 4^2. Here the left-hand side has k = 1 term, and the right-hand side has 2k = 2 terms. Hence k = 1 is in the sequence.
a(2): 29^2 + 30^2 + ... + 44^2 + 45^2 = 8^2 + 9^2 + ... + 40^2 + 41^2 = 23681. Here the left and right sums have k = 17 and 2k = 34 terms, respectively. Hence k = 17 is in the sequence.

Prog

(Python)
import sympy # Version 1.8
xx, yy = sympy.symbols("x y")
def pyramidal(n):
    return n*(n+1)*(2*n+1)/6 # A000330(n)
def expanded_diophantine(k, n):
    left_hand_side =  pyramidal(xx+n-1) - pyramidal(xx-1)
    right_hand_side =  pyramidal(yy+n+k-1) - pyramidal(yy-1)
    return sympy.expand(right_hand_side-left_hand_side)
def has_solutions(k, n):
    return len(sympy.solvers.diophantine(expanded_diophantine(k, n))) != 0
def k_in_a346163(k):
    return has_solutions(k, k)

Crossrefs

Cf. A000330, A001032, A130052.

Sequence in context: A275244 A002252 A064543 * A370853 A214791 A050712

Adjacent sequences: A346160 A346161 A346162 * A346164 A346165 A346166

Keyword

nonn

Author

Johan Westin, Jul 08 2021

Status

approved