A076114 a(n) = start of the smallest string of n consecutive positive integers with a square sum, or 0 if no such number exists.

1, 4, 2, 0, 3, 11, 4, 1, 5, 18, 6, 0, 7, 25, 8, 0, 9, 4, 10, 0, 11, 39, 12, 2, 4, 46, 14, 0, 15, 53, 16, 9, 17, 60, 18, 0, 19, 67, 20, 3, 21, 74, 22, 0, 23, 81, 24, 0, 1, 16, 26, 0, 27, 11, 28, 4, 29, 102, 30, 0, 31, 109, 32, 0, 33, 116, 34, 0, 35, 123, 36, 5, 37, 130, 11, 0, 39

Offset

1,2

Comments

Equivalently, smallest k such that n(n+2k-1)/2 is a square, 0 if there is no such square. - Ralf Stephan, Mar 23 2003

a(n) = 0 if and only if n has the form 4^e*m with e > 0 and m odd. - Dean Hickerson, Mar 25 2003

a(k) = 1 if k is the index of a square triangular number, i.e., k(k+1)/2 is a square, or if k belongs to A001108. If n is odd then a(n) <= (n+1)/2.

Links

Clement Martin, Table of n, a(n) for n = 1..10000

Example

a(2) = 4, 4+5 = 9 = 3^2.
a(8) = 1, 1+2+3+4+5+6+7+8 = 36 = 6^2.

Prog

(PARI) a(n) = if (!(n%4) && ((n/4^valuation(n, 4)) % 2), 0, my(o=n*(n+1)/2, k=0); while(!issquare(o+n*k), k++); k+1); \\ corrected by Michel Marcus, Sep 08 2025
(Python)
from itertools import count
from sympy.ntheory.primetest import is_square
def A076114(n):
    if not (n&1 or (~n & n-1).bit_length()&1): return 0
    return next(a for a in count(1) if is_square(n*(n+(a<<1)-1)>>1)) # Chai Wah Wu, Sep 16 2025

Crossrefs

Cf. A001108, A001109, A076115.

Sequence in context: A104689 A182501 A048819 * A294666 A051478 A121829

Adjacent sequences: A076111 A076112 A076113 * A076115 A076116 A076117

Keyword

nonn

Author

Amarnath Murthy, Oct 09 2002

Extensions

More terms from Ralf Stephan, Mar 23 2003

Status

approved