A190272 Numbers n such that n' = a -1, with n and a semiprimes and gcd(a,n) > 1, where n' is the arithmetic derivative of n.

6, 14, 15, 22, 33, 38, 46, 51, 62, 86, 87, 91, 95, 118, 141, 142, 145, 158, 159, 166, 206, 249, 262, 267, 278, 287, 295, 321, 326, 382, 395, 398, 411, 422, 445, 446, 473, 502, 519, 537, 542, 545, 581, 591, 622, 662, 695, 699, 703, 718, 745, 758, 766, 789, 838, 886, 895, 926, 951, 958, 995, 998, 1046, 1126, 1139, 1145, 1167, 1199, 1238, 1262, 1318, 1329, 1347, 1382, 1401, 1441, 1486, 1678, 1707, 1717, 1718, 1726, 1745, 1757, 1761, 1766

Offset: 1

Author

Giorgio Balzarotti, May 07 2011

Keywords

• nonn

Comments

This sequence is infinite, assuming Dickson's conjecture. In fact, the conjecture implies that there are infinitely many terms of this sequence divisible by any fixed prime p. - Charles R Greathouse IV, May 08 2011

Related to the Rassias Conjecture ("for any odd prime p there are primes q < r such that p*q = q+r+1") setting n = q*r, a = q+r+1. The sequence includes the cases with p = q (or p = r). Generalization can be achieved by removing the semiprimality condition or accepting gcd(n,a)=1. The differential equation in its general form n' = a + 1 includes Primary Pseudoperfect numbers, i.e., n' = n-1 (A054377).

Examples

			For n=6, 6' = 5, a = 5-1 = 4, gcd(4,6)=2, so 6 is a term.
		

Links

• For Rassias conjecture: Preda Mihăilescu, Review of Problem Solving and Selected Topics in Number Theory, Newsletter of the European Mathematical Society, March 2011, p. 46.

Crossrefs

Cf. A001358 (semiprimes), A003415 (arithmetic derivative), A054377 (Primary Pseudoperfect).

Programs

Maple

der:=n->n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
# for quick reference only
seq(`if`(bigomega(i)=2 and bigomega(der(i)+1)=2 and gcd(i,der(i)+1)>1,i,NULL),i=1..2000);

PARI

find(lim)=my(v=List());forprime(p=2,sqrtint(lim\2),forstep(q=2*p-1,lim\p,p+p,if(isprime(q\p+2)&isprime(q),listput(v,p*q))));vecsort(Vec(v)) \\ Charles R Greathouse IV, May 08 2011

Formula

Semiprimes pq with (p+q+1)/p prime. - Charles R Greathouse IV, May 08 2011