A287016 a(n) = smallest number k such that A071904(n) + k^2 is a perfect square.

0, 1, 2, 0, 3, 4, 1, 5, 2, 0, 7, 3, 8, 1, 4, 10, 5, 2, 0, 6, 13, 3, 14, 7, 1, 4, 17, 9, 2, 5, 0, 19, 10, 20, 6, 3, 22, 1, 12, 7, 4, 13, 25, 8, 2, 0, 5, 9, 28, 29, 16, 3, 6, 1, 32, 11, 18, 7, 4, 34, 19, 12, 35, 2, 0, 5, 21, 38, 9, 14, 3, 40, 6, 1, 15, 10, 24

Offset: 1

Author

Zhandos Mambetaliyev, Jean-François Alcover, May 18 2017

Keywords

• nonn
• easy

Examples

			The third odd composite number is A071904(3) = 21. and 21+2^2 = 25 = 5^2, so a(3) = 2.
		

Crossrefs

Cf. A016754, A037040, A071904.

Subsequence of A068527.

Programs

Mathematica

q[n_] := SelectFirst[Range[0, (n-1)/2], IntegerQ@ Sqrt[#^2 + n] &]; q /@ Select[Range[1, 300, 2], CompositeQ] (* Giovanni Resta, May 18 2017 *)

Python

from sympy import primepi, divisors
from sympy.ntheory.primetest import is_square
def A287016(n):
    if n == 1: return 0
    m, k = n, primepi(n) + n + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n + (k>>1)
    return 0 if is_square(int(m)) else -(d:=divisors(m))[l:=(len(d)>>1)-1]+d[l+1]>>1 # Chai Wah Wu, Aug 02 2024

Formula

a(m) = 0 for m>0 in A037040, the corresponding odd composites being in A016754\{1}. - Michel Marcus, May 19 2017

Extensions

More terms from Giovanni Resta, May 18 2017