A129385 a(n) is the smallest root m of the least perfect power q (= m^k) such that n+q is an even semiprime, or -1 if no such q exists.

2, 3, 2, 1, -1, 1, 2, 3, -1, 1, 2, 3, -1, 1, 2, 7, -1, 3, 2, 3, -1, 1, 2, 11, -1, 1, 2, 19, -1, 3, 2, 3, -1, 1, 2, 3, -1, 1, 2, 7, -1, 3, 2, 7, -1, 1, 2, 3, -1, 3, 2, 7, -1, 3, 2, 3, -1, 1, 2, 3, -1, 1, 2, 19, -1, 3, 2, 3, -1, 5, 2, 3, -1, 1, 2, 19, -1, 3, 2, 3, -1, 1, 2, 3, -1, 1, 2, 11, -1, 5, 2, 3, -1, 1, 2, 3, -1, 3, 2, 23, -1, 5, 2

Offset: 0

Author

Klaus Brockhaus, Apr 14 2007

Keywords

• sign

Comments

If n = 4*d with d > 0 then a(n) = -1: If q is odd then 4*d+q is odd; if q is even then q = 4*x with integer x > 0 and n+q = 2*2*(d+x) has more than 2 prime factors. Consequently n+q is odd or not semiprime.

There are also composite terms. The first two of them are a(122) = 6 and a(161) = 15.

Examples

			n=0: A001597(2) = 4 = 2^2 is the least perfect power q such that 0+q is an even semiprime; 0+4 = 4 = 2*2, hence a(0) = 2.
n=11: A001597(7) = 27 = 3^3 is the least perfect power q such that 11+q is an even semiprime; 11+27 = 38 = 2*19, hence a(11) = 3.
n=14: A001597(3) = 8 = 2^3 is the least perfect power q such that 14+q is an even semiprime; 14+8 = 22 = 2*11, hence a(14) = 2.
n=27: A001597(1722) = 2476099 = 19^5 is the least perfect power q such that 27+q is an even semiprime; 27+2476099 = 2476126 = 2*1238063 and 1238063 is prime, hence a(27) = 19.
		

Crossrefs

Cf. A001597 (perfect powers), A025478 (smallest root of perfect powers), A127913, A129386 (records), A129387 (where records occur).

Programs

Magma

PP:=[1] cat [ n: n in [2..2500000] | IsPower(n) ]; prootesp:=function(n); if exists(k) {x: x in PP | IsEven(n+x) and IsPrime((n+x) div 2) } then y:=k; else return -1; end if; if y eq 1 then return 1; end if; _, b:=IsPower(y); return b; end function; [ prootesp(n): n in [0..100] ];