A245639 -id:A245639 - cp's OEIS frontend

A245640 Prime numbers P such that 24*P^3-1 is also prime.

2, 3, 5, 7, 13, 17, 43, 61, 67, 73, 97, 127, 131, 137, 167, 241, 281, 307, 353, 433, 463, 467, 541, 557, 631, 641, 647, 653, 661, 673, 683, 821, 853, 857, 907, 911, 991, 1033, 1063, 1103, 1117, 1123, 1291, 1307, 1433, 1453, 1511, 1523, 1553, 1567, 1571, 1597, 1601, 1607

Offset: 1

Pierre CAMI, Jul 28 2014

			24*2^3-1=191 prime so a(1)=2.
24*3^3-1=647 prime so a(2)=3.

Pierre CAMI, Table of n, a(n) for n = 1..10000

Cf. A245639, A245641.

[p: p in PrimesUpTo(1700)| IsPrime(24*p^3-1)]; // Vincenzo Librandi, Sep 07 2014

Select[Prime[Range[300]], PrimeQ[24 #^3 - 1] &] (* Vincenzo Librandi, Sep 07 2014 *)

select(p->isprime(24*p^3-1), primes(300)) \\ Colin Barker, Jul 28 2014

A245641 Prime numbers P such that 8P^2-1 and 24P^3-1 are also primes.

Original entry on oeis.org

2, 3, 5, 17, 67, 137, 241, 353, 541, 641, 907, 1033, 1307, 1453, 1607, 1621, 1733, 1811, 2053, 2243, 2273, 2377, 2621, 2963, 3677, 3701, 3881, 3943, 4861, 5261, 5647, 6101, 6823, 7723, 7877, 8081, 8101, 8447, 8923, 9467, 10111, 10223, 11483, 11617, 12161, 12203, 12227, 12457

Offset: 1

Pierre CAMI, Jul 28 2014

nonn

Intersection of A245639 and A245640.

Pierre CAMI, Table of n, a(n) for n = 1..10000

Cf. A245639, A245640.

[p: p in PrimesUpTo(15000)| IsPrime(8*p^2-1)and IsPrime(24*p^3-1)]; // Vincenzo Librandi, Sep 08 2014

Select[Prime[Range[2000]], PrimeQ[8 #^2 - 1] && PrimeQ[24 #^3 - 1] &] (* Vincenzo Librandi, Sep 08 2014 *)

select(p->isprime(8*p^2-1)&&isprime(24*p^3-1), primes(3000)) \\ Colin Barker, Jul 28 2014

A245674 Prime numbers P such that 8P^2-1 and 8(8*P^2-1)^2-1 are also prime numbers.

Original entry on oeis.org

2, 79, 107, 173, 257, 359, 383, 523, 593, 971, 1493, 1811, 1867, 2273, 2357, 3187, 4111, 4723, 6389, 7607, 8101, 8699, 9473, 11027, 12157, 12227, 15017, 16301, 16987, 18797, 19801, 19913, 20071, 20323, 21313, 22003, 22307, 23203, 24229, 24733, 24859, 24943

Offset: 1

Pierre CAMI, Jul 29 2014

Subsequence of A245639.
For P < 10^9 in this sequence, 8*(8*(8*P^2-1)^2-1)^2-1 is composite.
Let f(x) = 8*x^2-1 and P >= 2. Then {f(P), f(f(P)), f(f(f(P)))} cannot all be prime. Proof: By doing cases on P mod 7, it can be shown that {f(P), f(f(P)), f(f(f(P)))} contains a multiple of 7. Also, all 3 numbers are greater than 7. - Jason Yuen, Feb 26 2025

			2 is prime, 8*2^2-1=31 is prime, 8*31^2-1=7687 is prime, so 2 is in the sequence.

Pierre CAMI, Table of n, a(n) for n = 1..10000

Cf. A245639.

[p: p in PrimesUpTo(25000)| IsPrime(8*p^2-1)and IsPrime(512*p^4-128*p^2+7)]; // Vincenzo Librandi, Sep 08 2014

f[n_]:=8 n^2 - 1; Select[Prime[Range[3000]], PrimeQ[f[#]]&&PrimeQ[f[f[#]]]&] (* Vincenzo Librandi, Sep 08 2014 *)

f(x) = 8*x^2-1;
forprime(p=1,10^5,if(ispseudoprime(f(p)) && ispseudoprime(f(f(p))), print1(p,", "))) \\ Derek Orr, Jul 29 2014

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A245640 Prime numbers P such that 24*P^3-1 is also prime.

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A245641 Prime numbers P such that 8P^2-1 and 24P^3-1 are also primes.

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A245674 Prime numbers P such that 8P^2-1 and 8(8*P^2-1)^2-1 are also prime numbers.

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cp's OEIS Frontend

A245640 Prime numbers P such that 24*P^3-1 is also prime.

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A245641 Prime numbers P such that 8*P^2-1 and 24*P^3-1 are also primes.

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A245674 Prime numbers P such that 8*P^2-1 and 8*(8*P^2-1)^2-1 are also prime numbers.

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A245641 Prime numbers P such that 8P^2-1 and 24P^3-1 are also primes.

A245674 Prime numbers P such that 8P^2-1 and 8(8*P^2-1)^2-1 are also prime numbers.