A126148 -id:A126148 - cp's OEIS frontend

A291464 Primes p such that p^3*q^3 + p^2 + q^2 is prime, where q is next prime after p.

2, 11, 13, 41, 97, 277, 389, 1093, 1229, 1409, 1429, 1627, 1823, 1931, 1979, 2437, 2521, 2549, 2657, 2689, 2719, 2729, 2731, 2969, 3019, 3413, 3539, 3593, 3613, 3623, 3697, 4003, 4027, 4289, 4327, 4583, 4751, 5051, 5323, 5503, 5657, 5783, 6143, 6221, 6299, 6329

Offset: 1

K. D. Bajpai, Aug 24 2017

nonn

			a(1) = 2 is prime; 3 is the next prime: 2^3*3^3 + 2^2 + 3^2 = 8*27 + 4 + 9 = 229 that is a prime.
a(2) = 11 is prime; 13 is the next prime: 11^3*13^3 + 11^2 + 13^2 = 1331*2197 + 121 + 169 = 2924497 that is a prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A001043, A006094, A030078, A096342, A120398, A126148, A152241, A291339, A291374.

[p: p in PrimesUpTo(5000) | IsPrime(p^3*q^3 + p^2 + q^2) where q is NextPrime(p)];

select(p -> andmap(isprime,[p,(p^3*nextprime(p)^3+p^2+nextprime(p)^2)]), [seq(p, p=1..10^4)]);

Select[Prime[Range[5000]], PrimeQ[#^3*NextPrime[#]^3 + #^2 + NextPrime[#]^2] &]
Select[Partition[Prime[Range[1000]],2,1],PrimeQ[#[[1]]^3 #[[2]]^3+#[[1]]^2+#[[2]]^2]&][[;;,1]] (* Harvey P. Dale, Sep 11 2023 *)

forprime(p=1, 5000, q=nextprime(p+1); p3=p^3; p2=p^2; q3=q^3; q2=q^2; if(ispseudoprime(p3*q3 + p2 + q2), print1(p, ", ")));

A358482 a(n) is the first prime p such that, if q is the next prime, (p*q+p+q)/5^n is a prime.

Original entry on oeis.org

2, 7, 1847, 90793, 139313, 1790293, 3834043, 5521543, 24996487, 2062865293, 5555052793, 12111965183, 95460776921, 6045070151921, 10204150316653, 70501997496487, 442748358250633, 368313674465183, 2935956099058987, 10360552690003447, 120999670013476223, 1820610211470152737

Offset: 0

Robert Israel, Dec 25 2022

nonn

Suggested in an email by J. M. Bergot.

For n >= 1, a(n) has the form k * 5^n + x, for some k >= 0, where x is a solution to the modular quadratic equation x^2 + (d+2)*x + d == 0 (mod 5^n), where d = q-p. - Daniel Suteu, Dec 28 2022

			a(2) = 1847 because 1847 is prime, the next prime is 1861, 1847*1861 + 1847 + 1861 = 3440975 = 5^2 * 137639 where 137639 is prime, and no smaller prime works.

Cf. A000351, A002386, A126148.

V:= Array(0..8):
q:= 2: count:= 0:
while count < 9 do
  p:= q; q:= nextprime(q);
t:= p*q+p+q;
k:= padic:-ordp(t,5);
if V[k] = 0 and isprime(t/5^k) then
  V[k]:= p; count:= count+1
fi
od:
convert(V,list);

a(11) from Michael S. Branicky, Dec 26 2022
a(12)-a(15) from David A. Corneth, Dec 26 2022
a(16) from Martin Ehrenstein, Dec 27 2022
a(17)-a(21) from Daniel Suteu, Dec 28 2022

cp's OEIS Frontend

A291464 Primes p such that p^3*q^3 + p^2 + q^2 is prime, where q is next prime after p.

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