cp's OEIS Frontend

A190275 Semiprimes of the form p*(p^2 - p + 1).

%I A190275 #26 Mar 07 2022 14:03:56
%S A190275 6,21,301,2041,296341,486877,2666437,3420301,4304341,7152001,38159521,
%T A190275 42387097,54296677,95235601,158048281,229971241,265434901,383712781,
%U A190275 454166017,775307917,972261181,1063290841,1304557801,1392422041,1730882401,1863895261,2631883561,2879450461,3714274297,3845297341,4070454361,4256780041,4849695001,5328809461,5722533337,5838483601,7218898681,7841065621
%N A190275 Semiprimes of the form p*(p^2 - p + 1).
%C A190275 This sequence is infinite, assuming Schinzel's Hypothesis H.
%C A190275 Related to Rassias Conjecture ("for any odd prime p there are primes q < r such that p*q = q + r + 1") setting p = q. Generalization can be achieved by removing semiprimality condition and accepting p^e, e >= 2.
%C A190275 These are semiprimes m = p*q such that 1/p + 1/q - 1/m = p/q. Cf. A326690. - _Amiram Eldar_ and _Thomas Ordowski_, Jul 22 2019
%H A190275 Giovanni Resta, <a href="/A190275/b190275.txt">Table of n, a(n) for n = 1..10000</a>
%H A190275 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 A190275 a(1) = 6 = 2*3 = 2*(2^2-2+1).
%e A190275 a(2) = 21 = 3*7 = 3*(3^2-3+1).
%e A190275 a(3) = 301 = 7*43 = 7*(7^2-7+1).
%p A190275 seq(`if`(isprime((ithprime(i)^2-ithprime(i)+1))=true,(ithprime(i)^2-ithprime(i)+1)*ithprime(i),NULL),i=1..300);
%t A190275 p = Select[Prime@ Range@ 500, PrimeQ[#^2 - # + 1] &]; p (p^2 - p + 1) (* _Giovanni Resta_, Jul 22 2019 *)
%o A190275 (PARI) forprime(p=2,1e4,if(isprime(k=p^2-p+1),print1(p*k", "))) \\ _Charles R Greathouse IV_, May 08 2011
%Y A190275 Cf. A065508 (primes p such that p^2-p+1 is prime).
%Y A190275 Cf. A001358 (semiprime), A003415 (arithmetic derivative), A164643, A190272 (n'=a-1), A190273 (n'=a+1), A190274 (n'=p^2-1).
%K A190275 nonn
%O A190275 1,1
%A A190275 _Giorgio Balzarotti_, May 07 2011