A159296 a(n) is the smaller number in the pair (L,m) which minimizes the primes of the form L^2 + m^2 under the constraint L + m = 2n + 1.

1, 2, 2, 4, 5, 5, 7, 7, 9, 8, 10, 12, 10, 14, 11, 14, 17, 15, 19, 18, 20, 22, 22, 24, 25, 25, 23, 26, 29, 30, 29, 32, 30, 34, 35, 34, 34, 37, 39, 31, 40, 42, 41, 40, 43, 44, 47, 45, 40, 50, 50, 47, 51, 52, 53, 55, 54, 56, 55, 60, 59, 61, 62, 55, 65, 65, 64, 66, 69, 70, 64, 72, 67, 72, 65

Offset: 1

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 09 2009

Keywords

• easy
• nonn

Comments

1) It is known that this sequence is infinite.

2) L and m with odd sum L + m are necessarily coprime if L^2 + M^2 is prime.

3) The "singular" case m = L = 1, L + m = 2 (even) with 1^2 + 1^2 = 2 is skipped. It would define a(0)=1.

4) a(n) <= n.

It has not been proved that a(n) exists for all n. See A036468. [T. D. Noe, Apr 22 2009]

Examples

			n=1: 1^2 + 2^2 = 5; a(1)=1.
n=2: 2^2 + 3^2 = 13 < 1^2 + 4^2 = 17; a(2)=2.
n=3: 2^2 + 5^2 = 29 < 1^2 + 6^2 = 37. 3^2 + 4^2 = 5^2 not prime; a(3)=2.
n=27: 23^2 + 32^2 = 1553 < 1597, 1657, 1693, 1733, 1777, 1877, 1933, 1993, 2273, 2437, 2617, 2713, 2917, a(27)=23.
		

Crossrefs

Cf. A145354, A157884.

Programs

Maple

A159296 := proc(n) local a,pmin,l,m ; a := 0 ; pmin := 2*(2*n+1)^2 ; for l from 1 to n do m := 2*n+1-l ; if isprime(m^2+l^2) then if m^2+l^2 < pmin then pmin := m^2+l^2 ; a := l ; fi; fi; od: RETURN(a) ; end: seq(A159296(n),n=1..80) ; # R. J. Mathar, Apr 18 2009

Extensions

Edited and extended by R. J. Mathar, Apr 18 2009