|
|
A363340
|
|
a(n) is the smallest positive integer such that a(n) * n is the sum of two squares.
|
|
4
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
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
|
|
LINKS
|
|
|
FORMULA
|
Multiplicative with a(p^e) = p if p^e == 3 (mod 4), otherwise 1. - Peter Munn, Jul 03 2023
(End)
|
|
EXAMPLE
|
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.
|
|
PROG
|
(PARI) a(n) = my(r=1); foreach(mattranspose(factor(n)), f, if(f[1]%4==3&&f[2]%2==1, r*=f[1])); r
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy,mult
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|