A129387 -id:A129387 - cp's OEIS frontend

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

Klaus Brockhaus, Apr 14 2007

sign

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.

			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.

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

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] ];

A129386 Records in A129385.

Original entry on oeis.org

2, 3, 7, 11, 19, 23, 43, 51, 57, 91, 119, 143, 167, 255, 263, 315, 355, 403, 475, 525, 579, 595, 611, 679, 691, 839

Offset: 1

Klaus Brockhaus, Apr 14 2007

			As can be gathered from A129385, the first four records are A129385(0) = 2, A129385(1) = 3, A129385(15) = 7, A129385(23) = 11. Hence a(1) to a(4) are 2, 3, 7, 11.

Cf. A129385, A129387 (where records occur).

lista(nn) = {my(m, q, r); for(k=2, nn, if(k%4 && !(k%2 && isprime((k+1)/2)), q=4+k%2; while(!ispower(q, , &m) || !isprime((q+k)/2), q+=2); if(m>r, print1(m, ", "); r=m))); } \\ Jinyuan Wang, Dec 04 2020

a(16) inserted by and a(19)-a(26) from Jinyuan Wang, Dec 04 2020

cp's OEIS Frontend

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.

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