A281543 Number of partitions n = x + y with y >= x > 0 such that x^2 + y^2 or (x^2 + y^2)/2 is prime.

0, 1, 1, 1, 2, 1, 2, 1, 2, 2, 3, 2, 3, 1, 4, 3, 4, 1, 4, 4, 3, 2, 4, 1, 8, 4, 4, 3, 6, 3, 5, 3, 4, 4, 9, 3, 8, 4, 6, 6, 9, 2, 7, 4, 7, 5, 7, 3, 5, 7, 7, 6, 9, 4, 14, 4, 8, 4, 9, 4, 11, 7, 7, 6, 17, 5, 11, 6, 10, 8, 9, 5, 11, 6, 9, 7, 8, 3, 13, 9, 9, 5, 15, 5, 20, 8, 11, 8, 14, 7, 13, 9, 8, 6, 18, 7, 14, 10, 10, 8

Offset: 1

Author

Thomas Ordowski and Altug Alkan, Mar 01 2017

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 1.

We have a(n) <= phi(n)/2 for n <> 2, because must be gcd(x,y) = 1.

Numbers n such that a(n) = phi(n)/2 are 3, 4, 5, 6, 10, 12, 15, and 20.

Record values of a(n) are for n = 1, 2, 5, 11, 15, 25, 35, 55, 65, 85, 125, 145, 185, 205, 215, 235, 265, 295, 325, 365, 415, ... cf. A001750.

Examples

			a(5) = 2 because 5 = 1 + 4 and 5 = 2 + 3 are only options; 1^2 + 4^2 = 17 and 2^2 + 3^2 = 13 are primes.
a(6) = 1 because 6 = 1 + 5 is only option; (1^2 + 5^2)/2 = 13 is prime.
a(7) = 2 because 7 = 1 + 6, 7 = 2 + 5 and 7 = 3 + 4, but 3^2 + 4^2 = 5^2.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Altug Alkan, Alternative Scatterplot of A281543

Crossrefs

Cf. A000010, A002313, A036468, A069004.

Programs

PARI

a(n) = if(n==2, 1, if(n%2==0, sum(k=1, n/2-1, isprime(n^2/4+k^2)), sum(k=1, (n-1)/2, isprime(k^2+(n-k)^2))));

Formula

a(2m+1) = A036468(m) for m > 0.

a(2m) = A069004(m) for m > 1.

a(n) = O(n/log(n)).