A328022 - OEIS

%I #31 Oct 14 2019 12:19:12

%S 17,41,367,514275529

%N 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 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 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 a(5) > 10^22, if it exists. - _Giovanni Resta_, Oct 02 2019

%e Known values:

%e    n |  a(n) = p =        i *  q + r

%e   ===+==============================

%e    1 |        17 =        7 *  2 + 3

%e    2 |        41 =       13 *  3 + 2

%e    3 |       367 =       73 *  5 + 2

%e    4 | 514275529 = 27067133 * 19 + 2

%t Select[Prime@ Range[10^5], AllTrue[Join[{#1, #2}, QuotientRemainder[#1, #2]], PrimeQ] & @@ {#, PrimePi@ #} &] (* _Michael De Vlieger_, Oct 01 2019 *)

%o (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 Cf. A023144, A156152, A156153.

%K nonn,hard,more

%O 1,1

%A _Eduardo P. Feitosa_, Oct 01 2019

%E a(4) from _Andrew Howroyd_, Oct 01 2019