cp's OEIS Frontend

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.

%I A190272 #19 Aug 18 2019 16:45:07
%S A190272 6,14,15,22,33,38,46,51,62,86,87,91,95,118,141,142,145,158,159,166,
%T A190272 206,249,262,267,278,287,295,321,326,382,395,398,411,422,445,446,473,
%U A190272 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
%N 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.
%C A190272 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
%C A190272 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).
%H A190272 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.
%F A190272 Semiprimes pq with (p+q+1)/p prime. - _Charles R Greathouse IV_, May 08 2011
%e A190272 For n=6, 6' = 5, a = 5-1 = 4, gcd(4,6)=2, so 6 is a term.
%p A190272 der:=n->n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
%p A190272 # for quick reference only
%p A190272 seq(`if`(bigomega(i)=2 and bigomega(der(i)+1)=2 and gcd(i,der(i)+1)>1,i,NULL),i=1..2000);
%o A190272 (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
%Y A190272 Cf. A001358 (semiprimes), A003415 (arithmetic derivative), A054377 (Primary Pseudoperfect).
%K A190272 nonn
%O A190272 1,1
%A A190272 _Giorgio Balzarotti_, May 07 2011