cp's OEIS Frontend

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.

%I A190273 #18 Jul 27 2019 16:53:56
%S A190273 6,10,21,26,39,55,57,74,93,111,122,146,155,201,203,253,301,305,314,
%T A190273 327,381,386,417,471,497,543,554,597,626,633,689,737,755,791,794,842,
%U A190273 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
%N 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.
%C A190273 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).
%C A190273 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
%H A190273 For Rassias conjecture: Preda Mihăilescu, <a href="http://www.ems-ph.org/journals/newsletter/pdf/2011-03-79.pdf">Review of Problem Solving and Selected Topics in Number Theory</a>, Newsletter of the European Mathematical Society, March 2011, p. 46.
%e A190273 n=6, 6'=5, m=5+1=6, gcd(6,6)=6 -> a(1)=6
%p A190273 der:=n->n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
%p A190273 seq(`if`(bigomega(i)=2 and bigomega(der(i)-1)=2 and gcd(i,der(i)-1)>1,i,NULL),i=1..2000);
%Y A190273 Cf. A001358 (semiprimes), A003415 (arithmetic derivative), A007850 (Giuga numbers), A190272 (n'=m-1), A190273, A190274, A190275.
%K A190273 nonn
%O A190273 1,1
%A A190273 _Giorgio Balzarotti_, May 07 2011