A109953 -id:A109953 - cp's OEIS frontend

A241099 Primes p such that (p^3 + 4)/3 is prime.

5, 23, 53, 113, 173, 197, 269, 317, 383, 443, 557, 563, 587, 647, 659, 773, 797, 827, 947, 983, 1097, 1103, 1187, 1217, 1229, 1889, 1913, 1949, 2039, 2099, 2153, 2213, 2339, 2357, 2399, 2417, 2447, 2579, 2693, 2837, 2879, 2897, 2903, 2939, 2969, 3089, 3203

Offset: 1

K. D. Bajpai, Apr 15 2014

nonn

			5 is prime and appears in the sequence because (5^3 + 4)/3 = 43 which is a prime.
23 is prime and appears in the sequence because (23^3 + 4)/3 = 4057 which is a prime.

K. D. Bajpai, Table of n, a(n) for n = 1..8857

Cf. A109953 (primes p:(p^2+1)/3 is prime).
Cf. A118915 (primes p:(p^2+5)/6 is prime).
Cf. A118918 (primes p:(p^2+11)/12 is prime).

KD:= proc() local a,b;a:=ithprime(n); b:=(a^3+4)/3; if b=floor(b) and isprime(b) then RETURN (a); fi; end: seq(KD(), n=1..1000);

Select[Prime[Range[500]], PrimeQ[(#^3 + 4)/3] &]
n = 0; Do[If[PrimeQ[(Prime[k]^3 + 4)/3], n = n + 1; Print[n, " ", Prime[k]]], {k, 1, 200000}] (* b-file *)

A245590 Primes p such that p^2 + 6 is a semiprime.

Original entry on oeis.org

2, 3, 7, 17, 23, 41, 47, 53, 59, 101, 149, 157, 173, 179, 193, 211, 229, 233, 239, 241, 251, 311, 347, 349, 353, 359, 373, 379, 383, 389, 409, 421, 439, 443, 457, 479, 499, 509, 521, 541, 571, 577, 599, 619, 641, 661, 691, 701, 719, 751, 761, 769, 809, 823, 829

Offset: 1

K. D. Bajpai, Jul 26 2014

			7 is in the sequence because it is prime and 7^2 + 6 = 55 = 5 * 11, which is semiprime.
23 is in the sequence because it is prime and 23^2 + 6 = 535 = 5 * 107, which is semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..11200

Cf. A000040, A001358.
Cf. A109953 (primes p: p^2 + 2 is semiprime).
Cf. A243365 (primes p: p^2 + 6 and p^2 - 6  are semiprimes).

with(numtheory):A245590:=n->`if`(isprime(n) and bigomega(n^2+6)=2, n, NULL): seq(A245590 (n), n=1..1500);

Select[Prime[Range[200]], PrimeOmega[#^2 + 6] == 2 &]

forprime(p=1,10^4,if(bigomega(p^2+6)==2,print1(p,", "))) \\ Derek Orr, Aug 03 2014

A351051 a(n) is the least prime that begins a sequence of exactly n primes under iteration of the map x -> (x^2+2)/3.

Original entry on oeis.org

3, 11, 17, 7, 25781659, 13505561767

Offset: 1

J. M. Bergot and Robert Israel, Jan 30 2022

			7 is prime, (7^2+2)/3 = 17 is prime, (17^2+2)/3 = 97 is prime, (97^2+2)/3 = 3137 is prime, but (3137^2+2)/3 = 3280257 is not prime, so 7 begins the sequence of 4 primes (7, 17, 97, 3137).  Since this is the first prime to do so, a(4) = 7.

Cf. A109953.

f:= proc(p) option remember; local q;
  q:= (p^2+2)/3;
  if isprime(q) then 1 + procname(q) else 1 fi
end proc:
A:= Vector(5): count:= 0:
p:= 3:
while count < 5 do
p:= nextprime(p);
v:= f(p);
if A[v] = 0 then A[v]:= p; count:= count+1; fi;
od:
convert(A,list);

f[n_] := -1 + Length @ NestWhileList[(#^2 + 2)/3 &, n, PrimeQ]; a[n_] := Module[{p = 3}, While[f[p] != n, p = NextPrime[p]]; p]; Array[a, 4] (* Amiram Eldar, Feb 01 2022 *)

a(6) from Amiram Eldar, Feb 01 2022

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A241099 Primes p such that (p^3 + 4)/3 is prime.

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A245590 Primes p such that p^2 + 6 is a semiprime.

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A351051 a(n) is the least prime that begins a sequence of exactly n primes under iteration of the map x -> (x^2+2)/3.

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