A190273 Numbers n such that n' = m+1, with n and m semiprimes and gcd(m,n)>1, where n' is the arithmetic derivative of n.

6, 10, 21, 26, 39, 55, 57, 74, 93, 111, 122, 146, 155, 201, 203, 253, 301, 305, 314, 327, 381, 386, 417, 471, 497, 543, 554, 597, 626, 633, 689, 737, 755, 791, 794, 842, 889, 905, 914, 921, 1011, 1027, 1055, 1081, 1082, 1137, 1226, 1227, 1322, 1346, 1379, 1461, 1466, 1477, 1497, 1514, 1623, 1655, 1703, 1711, 1713, 1731, 1751, 1754, 1893, 1967, 1994

Offset: 1

Author

Giorgio Balzarotti, May 07 2011

Keywords

• nonn

Comments

The sequence is related to the Rassias Conjecture ("for any prime p there are two primes p1 and p2 such that p*p1=p1+p2+1, p>2, p2>p1", see A190272-A190275), because n = p1*p2, m=p1*p -> p1*p = p1+p2-1. The sequence includes the cases with p=p1 (or p2). Generalization can be achieved by removing semiprimarity condition or accepting gcd(n,m)=1. The differential equation in its general form n'=m+1 includes Giuga Numbers, i.e., n'=b*n+1, or n'=n+1 (A007850).

These are semiprimes n = p*q such that 1/p + 1/q - 1/n = P/Q, where P <> Q are primes. Cf. A326690. - Amiram Eldar and Thomas Ordowski, Jul 25 2019

Examples

			n=6, 6'=5, m=5+1=6, gcd(6,6)=6 -> a(1)=6
		

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), A007850 (Giuga numbers), A190272 (n'=m-1), A190273, A190274, A190275.

Programs

Maple

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