cp's OEIS Frontend

A328022 Prime numbers p such that all 4 variables of the equation (p = i * q + r) are prime, with i being the index of p, q the quotient of p/i, and r the remainder of p/i.

%I A328022 #31 Oct 14 2019 12:19:12
%S A328022 17,41,367,514275529
%N A328022 Prime numbers p such that all 4 variables of the equation (p = i * q + r) are prime, with i being the index of p, q the quotient of p/i, and r the remainder of p/i.
%C A328022 The other two variables in the equation result from the division of a prime p by its index i, giving quotient q and remainder r. All four of p, i, q, r are required to be prime.
%C A328022 For all remaining terms, q (which has become greater than 2) will be an odd prime, and q increases exponentially slowly. And when q is odd, exactly one of i and r will be odd. Consequently, a new term will only occur when r = 2 and both q and i are prime.
%C A328022 a(5) > 10^22, if it exists. - _Giovanni Resta_, Oct 02 2019
%e A328022 Known values:
%e A328022    n |  a(n) = p =        i *  q + r
%e A328022   ===+==============================
%e A328022    1 |        17 =        7 *  2 + 3
%e A328022    2 |        41 =       13 *  3 + 2
%e A328022    3 |       367 =       73 *  5 + 2
%e A328022    4 | 514275529 = 27067133 * 19 + 2
%t A328022 Select[Prime@ Range[10^5], AllTrue[Join[{#1, #2}, QuotientRemainder[#1, #2]], PrimeQ] & @@ {#, PrimePi@ #} &] (* _Michael De Vlieger_, Oct 01 2019 *)
%o A328022 (PARI) lista(nn)={my(i=1); forprime(p=3, nn, i++; if(isprime(i), my(q=p\i); if(isprime(q)&&isprime(p-q*i), print1(p, ", ")) ))} \\ _Andrew Howroyd_, Oct 01 2019
%Y A328022 Cf. A023144, A156152, A156153.
%K A328022 nonn,hard,more
%O A328022 1,1
%A A328022 _Eduardo P. Feitosa_, Oct 01 2019
%E A328022 a(4) from _Andrew Howroyd_, Oct 01 2019