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.

17, 41, 367, 514275529

Offset: 1

Author

Eduardo P. Feitosa, Oct 01 2019

Keywords

• nonn
• hard
• more

Comments

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.

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.

a(5) > 10^22, if it exists. - Giovanni Resta, Oct 02 2019

Examples

			Known values:
   n |  a(n) = p =        i *  q + r
  ===+==============================
   1 |        17 =        7 *  2 + 3
   2 |        41 =       13 *  3 + 2
   3 |       367 =       73 *  5 + 2
   4 | 514275529 = 27067133 * 19 + 2
		

Crossrefs

Cf. A023144, A156152, A156153.

Programs

Mathematica

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

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

Extensions

a(4) from Andrew Howroyd, Oct 01 2019