A156152 -id:A156152 - cp's OEIS frontend

A156153 Primes p such that p == 2 (mod pi(p)) and pi(p) is prime.

5, 41, 367, 9559817, 514275529

Offset: 1

M. F. Hasler, Feb 04 2009

p=c=0; until(0, until( isprime(c++), p=nextprime(p+1)); (p-2)%c & next; print1(p, ", ")); \\ Modified by Jinyuan Wang, Feb 22 2020

a(n) = A000040(A156154(n)) = A000040(A000040(A156155(n))).

A156151 Primes p such that p+2 = 0 (mod pi(p)), where pi(p)=A000720(p) is the prime counting function.

Original entry on oeis.org

2, 31, 353, 9559783, 9559843, 9559903, 3779853313, 27788573801, 204475054073, 204475054723, 1505578024807, 1505578025779, 241849345578351691, 1784546064357413809, 1784546064357419959, 97199410027249994623, 97199410027250046643, 97199410027250047453, 97199410027250123143

Offset: 1

M. F. Hasler, Feb 04 2009

nonn

Cf. A156152.

p=c=0; until(0, (2+p=nextprime(p+1))%c++ || print1(p",")) \\ PARI syntax for || updated Feb 20 2020

a(n) = A000040(A092044(n)).

More terms from Max Alekseyev, May 03 2009

a(13)-a(19) from Giovanni Resta, Feb 23 2020

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.

Original entry on oeis.org

17, 41, 367, 514275529

Offset: 1

Eduardo P. Feitosa, Oct 01 2019

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

			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

Cf. A023144, A156152, A156153.

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

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

a(4) from Andrew Howroyd, Oct 01 2019

cp's OEIS Frontend

A156153 Primes p such that p == 2 (mod pi(p)) and pi(p) is prime.

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A156151 Primes p such that p+2 = 0 (mod pi(p)), where pi(p)=A000720(p) is the prime counting function.

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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.

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