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

1,3

Links

Table of n, a(n) for n=1..77.

Formula

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A016754, A037040, A071904.

Subsequence of A068527.

Sequence in context: A066439 A352579 A241319 * A368312 A285722 A274441

Adjacent sequences: A287013 A287014 A287015 * A287017 A287018 A287019

Keyword

nonn,easy

Author

Zhandos Mambetaliyev, Jean-François Alcover, May 18 2017

Extensions

More terms from Giovanni Resta, May 18 2017

Status

approved