A363340 a(n) is the smallest positive integer such that a(n) * n is the sum of two squares.

1, 1, 3, 1, 1, 3, 7, 1, 1, 1, 11, 3, 1, 7, 3, 1, 1, 1, 19, 1, 21, 11, 23, 3, 1, 1, 3, 7, 1, 3, 31, 1, 33, 1, 7, 1, 1, 19, 3, 1, 1, 21, 43, 11, 1, 23, 47, 3, 1, 1, 3, 1, 1, 3, 11, 7, 57, 1, 59, 3, 1, 31, 7, 1, 1, 33, 67, 1, 69, 7, 71, 1, 1, 1, 3, 19, 77, 3, 79

Offset: 1

Author

Peter Schorn, May 28 2023

Keywords

• nonn
• easy
• mult

Comments

Using Fermat's two-squares theorem it is easy to see that a(n) is the product of all prime factors of n that are congruent to 3 modulo 4 and have an odd exponent.

This implies that a(n) is also the smallest positive integer such that n / a(n) is the sum of two squares.

Equivalently, a(n) is the product of all primes of the form 4k+3 that divide the squarefree part of n. If we use the squarefree kernel instead, we get A170819. - Peter Munn, Aug 06 2023

Examples

			a(1) = a(2) = 1 since 1 and 2 are sums of two squares.
a(3) = 3 since 3 and 6 are not sums of two squares but 3*3 is.
a(6) = 3 since 6 and 12 are not sums of two squares but 3*6 = 3^2 + 3^2.
		

Crossrefs

Cf. A001481 (positions of 1's), A167181 (range of values).

Fixed points: A167181.

Cf. A000265, A002145, A007913, A059897, A097706, A170819.

Programs

PARI

a(n) = my(r=1); foreach(mattranspose(factor(n)), f, if(f[1]%4==3&&f[2]%2==1, r*=f[1])); r

Formula

Multiplicative with a(p^e) = p if p^e == 3 (mod 4), otherwise 1. - Peter Munn, Jul 03 2023

From Peter Munn, Aug 06 2023: (Start)

a(n) = A007913(A097706(n)) = A097706(A007913(n)).

a(n) == A000265(n) (mod 4).

a(A059897(n, k)) = A059897(a(n), a(k)).

(End)