A106483 -id:A106483 - cp's OEIS frontend

A224990 Primes p such that q = 2p^2 - 1 and 2p*q - 1 are also prime.

3, 13, 157, 181, 739, 829, 937, 1009, 1093, 1483, 1621, 1879, 2311, 2503, 2647, 2719, 3079, 4969, 4999, 5209, 5431, 5569, 6163, 6961, 8161, 8329, 9349, 9631, 10399, 10459, 10531, 10657, 11131, 11953, 13063, 18523, 20149, 20731, 21391, 21589, 26317, 27481, 28111, 28351, 29023

Offset: 1

M. F. Hasler, Apr 22 2013

nonn

Subsequence of A106483, and more elementary version of A224614.

Select[Prime[Range[10000]], PrimeQ[2#^2 - 1] && PrimeQ[4#^3 - 2# - 1] &] (* Alonso del Arte, Apr 22 2013 *)

forprime(p=1,3e4,isprime(r=2*p^2-1)&&isprime(2*p*r-1)&&print1(p","))

A245659 Prime numbers P such that Q=2P^2-1, R=2Q^2-1, S=2R^2-1 and T=2S^2-1 are all prime numbers.

Original entry on oeis.org

281683, 496789, 823421, 1352753, 1719217, 6174109, 8643149, 9761051, 9843529, 16191167, 19132121, 19745797, 23490473, 28457797, 31820429, 32860271, 36552277, 37068569, 43506569, 44776981, 46808903, 55035047, 55957807, 67194403, 75099137, 83092897, 86580421, 89135089

Offset: 1

Pierre CAMI, Jul 28 2014

nonn

Subsequence of A106483.

For P = 496789, 83092897, 467014643, U=2*T^2-1 is also prime. [Corrected by Jens Kruse Andersen, Aug 21 2014]

			281683 is prime P.
Q=2*P^2-1 = 158690624977 is prime Q.
R=2*Q^2-1 = 50365428911181712501057 is prime R.
S=2*R^2-1 = 5073352858814597404058971422301788780452234497 is prime S.
T=2*S^2-1 = 51477818460084496601334991724899650493354568309112026195311592373475872924903206720553686017 is prime T.
U=2*T^2-1 is composite.

Pierre CAMI, Table of n, a(n) for n = 1..140

Cf. A106483.

f[n_]:=2n^2-1;Select[Prime[Range[5170000]],PrimeQ[f[#]]&&PrimeQ[ f[f[#]]]&&PrimeQ[ f[f[f[#]]]]&&PrimeQ[f[f[f[f[#]]]]]&] (* Farideh Firoozbakht, Aug 11 2014 *)
Select[Prime[Range[52*10^5]],AllTrue[Rest[FoldList[2#^2-1&,{#,#,#,#,#}]],PrimeQ]&] (* Harvey P. Dale, Jan 13 2023 *)

f(x)=return(2*x^2-1)
forprime(p=1,10^8,if(ispseudoprime(f(p)) && ispseudoprime(f(f(p))) && ispseudoprime(f(f(f(p)))) && ispseudoprime(f(f(f(f(p))))), print1(p,", "))) \\ Derek Orr, Jul 28 2014

More terms from Derek Orr, Jul 28 2014

A340865 Primes p such that (p^2 + 1)/2 and 2*p^2 - 1 are also prime.

Original entry on oeis.org

3, 11, 59, 181, 199, 379, 409, 571, 739, 1039, 1439, 2239, 2269, 2351, 2381, 2671, 2719, 2789, 3049, 3529, 4021, 4201, 4721, 4999, 5431, 5531, 5839, 6329, 6619, 8329, 9241, 9419, 9631, 9689, 10151, 11329, 11551, 12071, 12421, 13339, 14489, 15091, 17419, 18301

Offset: 1

Jon E. Schoenfield, Jan 24 2021

nonn

Intersection of A048161 and A106483.
How many triangular numbers with 6 divisors (A292989) can be divisible by the same squared prime p^2?
The k-th triangular number T(k) = A000217(k) = k*(k+1)/2 can be written as the product of two coprime factors A and B where A=k and B=(k+1)/2 for odd k, A=k/2 and B=k+1 for even k. If a triangular number has 6 divisors, then it is of the form p^2*q where p and q are distinct primes. We can identify four cases:
Case 1: A = k = p^2 and B = (k+1)/2 = q, so q = (p^2 + 1)/2; solutions occur at primes p in A048161.
Case 2: A = k = q and B = (k+1)/2 = p^2, so 2*p^2 - 1 = q; solutions occur at primes p in A106483.
Case 3: A = k/2 = p^2 and B = k+1 = q. In this case, 2*p^2 + 1 = q. For p = 2, we would get q = 9 (nonprime), so p must be odd. If prime p > 3 (so q > 19), we have p^2 == 1 (mod 3), so q == 0 (mod 3), hence nonprime. So the only solution for this case occurs at p=3, q=19, t = 3^2*19 = 171.
Case 4: A = k/2 = q and B = k+1 = p^2. In this case, 2*q + 1 = p^2, so p is odd, but then p^2 == 1 (mod 8), so q == 0 (mod 4), hence q is not prime: no solutions exist.
Since Case 4 has no solutions, at most three triangular numbers with 6 divisors can be divisible by the same squared prime p^2; Case 3 has a solution only at p=3 and, in fact, there are three triangular numbers with 6 divisors that are divisible by 3^2: t = 3^2*5 = 45 = T(9), t = 3^2*17 = 153 = T(17), and 3^2*19 = 171 = T(18).
For all primes p > 3, then, at most two triangular numbers with 6 divisors are divisible by p^2; this sequence (after the initial term, 3) lists the primes p such that p^2 divides exactly two triangular numbers that have 6 divisors.

			Both (3^2 + 1)/2 = 5 and 2*3^2 - 1 = 17 are prime, so 3 is in the sequence.
(5^2 + 1)/2 = 13 is prime, but 2*5^2 - 1 = 49 = 7^2 is not prime, so 5 is not in the sequence.
(7^2 + 1)/2 = 25 is not prime, so even though 2*7^2 - 1 = 97 is prime, 7 is not in the sequence.
Neither (23^2 + 1)/2 = 265 = 5*53 nor 2*23^2 - 1 = 1057 = 7*151 is prime, so 23 is not in the sequence.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Cf. A000217, A048161, A106483, A292989.

isok(p) = (p>2) && isprime(p) && isprime((p^2+1)/2) && isprime(2*p^2-1); \\ Michel Marcus, Jan 25 2021

A356510 Primes p such that 2p^2 - 7, 2p^2 - 1, and 2*p^2 + 3 are prime.

Original entry on oeis.org

43, 127, 197, 3613, 3767, 4957, 28687, 29723, 40193, 46817, 66403, 78737, 89137, 93253, 104243, 105337, 105673, 110543, 114113, 123397, 127247, 145963, 148303, 168713, 173293, 190387, 201893, 207367, 213613, 241597, 256117, 261323, 268253, 278543, 283807, 333227, 339373, 340913, 356173, 359143

Offset: 1

J. M. Bergot and Robert Israel, Aug 09 2022

nonn

All terms == 3 or 7 (mod 10).

All terms == 1 or 13 (mod 14). - Jon E. Schoenfield, Sep 05 2022

			a(3) = 197 is a term because 197, 2*197^2 - 7 = 77611, 2*197^2 - 1 = 77617, and 2*197^2 + 3 = 77621 are all prime.

Robert Israel, Table of n, a(n) for n = 1..1000

Contained in A106483 and A243595.

filter:= p -> isprime(p) and isprime(2*p^2+3) and isprime(2*p^2-1) and isprime(2*p^2-7):
select(filter, [seq(i,i=3..1000000,2)]);

Select[Prime[Range[30000]], AllTrue[2*#^2 + {-7, -1, 3}, PrimeQ] &] (* Amiram Eldar, Aug 09 2022 *)

from sympy import isprime
def ok(n): return isprime(n) and all(isprime(2*n*n-i) for i in [7, 1, -3])
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Aug 09 2022

A213162 Smallest prime p such that (k+1)*p^2-k are prime for k=1..n.

Original entry on oeis.org

2, 199, 409, 17569, 981091, 3918251, 1329433951, 75902670689, 45048280453021

Offset: 1

Zak Seidov, Jun 06 2012

a(1..8) are the first terms correspondingly in A106483, A213078, A213079, A213107, A213125, A213159, A213161, A213334.

a(n) = {my(p = 2); until (ok, ok = 1; for (k = 1, n, if (! isprime((k+1)*p^2-k), ok = 0; break;);); if (!ok, p = nextprime(p+1);)); return (p);}  \\ Michel Marcus, Apr 19 2013

a(9) from Tyler Busby, Jan 11 2023

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A224990 Primes p such that q = 2*p^2 - 1 and 2*p*q - 1 are also prime.

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A245659 Prime numbers P such that Q=2*P^2-1, R=2*Q^2-1, S=2*R^2-1 and T=2*S^2-1 are all prime numbers.

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A340865 Primes p such that (p^2 + 1)/2 and 2*p^2 - 1 are also prime.

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A356510 Primes p such that 2*p^2 - 7, 2*p^2 - 1, and 2*p^2 + 3 are prime.

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A213162 Smallest prime p such that (k+1)*p^2-k are prime for k=1..n.

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A224990 Primes p such that q = 2p^2 - 1 and 2p*q - 1 are also prime.

A245659 Prime numbers P such that Q=2P^2-1, R=2Q^2-1, S=2R^2-1 and T=2S^2-1 are all prime numbers.

A356510 Primes p such that 2p^2 - 7, 2p^2 - 1, and 2*p^2 + 3 are prime.