A049002 -id:A049002 - cp's OEIS frontend

A062326 Primes p such that p^2 - 2 is also prime.

2, 3, 5, 7, 13, 19, 29, 37, 43, 47, 61, 71, 89, 103, 107, 127, 131, 139, 173, 191, 211, 223, 233, 239, 257, 293, 313, 337, 359, 421, 443, 449, 467, 491, 523, 541, 569, 587, 607, 653, 677, 719, 727, 733, 743, 751, 757, 761, 797, 811, 863, 881, 1013, 1021

Offset: 1

Reiner Martin, Jul 12 2001

When p and p^2 - 2 are both prime, the fundamental solution of the Pell equation x^2 - n*y^2 = 1, for n = p^2 - 2, is x = p^2 - 1 and y = p. See A109748 for the case of n and x both prime. - T. D. Noe, May 19 2007
3 is the only prime p such that p^2 + 2 and p^2 - 2 are both primes. - Jaroslav Krizek, Nov 25 2013 (note that p^2 + 2 is composite for all primes p >= 5. - Joerg Arndt, Jan 10 2015)
For all primes p except for p = 3, p^2 + 2 is multiple of 3 (see A061725). - Zak Seidov, Feb 19 2015

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A049002 (p^2-2).

Cf. A137291, A010051, A004901, A000040.

import Data.List (elemIndices)
a062326 = a000040 . a137291
a062326_list = map (a000040 . (+ 1)) $
               elemIndices 1 $ map a010051' a049001_list
-- Reinhard Zumkeller, Jul 30 2015

[ p: p in PrimesUpTo(1100) | IsPrime(p^2-2) ]; // Klaus Brockhaus, Jan 01 2009

Select[Prime[Range[500]], PrimeQ[#^2 - 2] &] (* Harvey P. Dale, Sep 20 2011 *)

{ n=0; forprime (p=2, 5*10^5, if (isprime(p^2 - 2), write("b062326.txt", n++, " ", p); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 05 2009

A063637 Primes p such that p+2 is a semiprime.

Original entry on oeis.org

2, 7, 13, 19, 23, 31, 37, 47, 53, 67, 83, 89, 109, 113, 127, 131, 139, 157, 167, 181, 199, 211, 233, 251, 257, 263, 293, 307, 317, 337, 353, 359, 379, 389, 401, 409, 443, 449, 467, 479, 487, 491, 499, 503, 509, 541, 557, 563, 571, 577, 587, 631, 647, 653, 677

Offset: 1

Reinhard Zumkeller, Jul 21 2001

nonn

Primes of the form p*q - 2, where p and q are primes.

Union of A049002 and A115093. - T. D. Noe, Mar 01 2006

			From _K. D. Bajpai_, Sep 06 2014: (Start)
a(3) = 13, which is prime, and 13 + 2 = 15 = 3 * 5, which is a semiprime.
a(4) = 19, which is prime, and 19 + 2 = 21 = 3 * 7, which is a semiprime.
(End)

J.-R. Chen, On the representation of a large even integer as the sum of a prime and a product of at most two primes, Sci. Sinica 16 (1973), 157-176.

K. D. Bajpai, Table of n, a(n) for n = 1..14190 (first 1000 terms from T. D. Noe)
P. Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 146. [?Broken link]
P. Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 146.
T. Tao, Obstructions to uniformity and arithmetic patterns in the primes, arXiv:math/0505402 [math.NT], 2005.

Cf. A005383, A001358, A063638.

Cf. A109611 (Chen primes).

a063637 n = a063637_list !!(n-1)
a063637_list = filter ((== 1) . a064911 . (+ 2)) a000040_list
-- Reinhard Zumkeller, Nov 15 2011

select(t -> isprime(t) and numtheory:-bigomega(t+2)=2, [2, seq(2*i+1,i=1..500)]); # Robert Israel, Sep 07 2014

f[n_] := Plus @@ Flatten[ Table[ # [[2]], {1}] & /@ FactorInteger[ n]]; Select[ Prime[ Range[ 123]], f[ # + 2] == 2 &] (* Robert G. Wilson v, Apr 30 2005 *)
Select[Prime[Range[500]],PrimeOmega[#+2]==2&]  (* K. D. Bajpai, Sep 06 2014 *)

{ n=0; for (m=1, 10^9, p=prime(m); if (bigomega(p + 2) == 2, write("b063637.txt", n++, " ", p); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 26 2009

a(n) = A062721(n) - 2.

A010051(a(n)) * A064911(a(n) + 2) = 1. - Reinhard Zumkeller, Nov 15 2011

A049001 a(n) = prime(n)^2 - 2.

Original entry on oeis.org

2, 7, 23, 47, 119, 167, 287, 359, 527, 839, 959, 1367, 1679, 1847, 2207, 2807, 3479, 3719, 4487, 5039, 5327, 6239, 6887, 7919, 9407, 10199, 10607, 11447, 11879, 12767, 16127, 17159, 18767, 19319, 22199, 22799, 24647, 26567, 27887

Offset: 1

N. J. A. Sloane

Smallest numbers k such that k*prime(n)^2 + 1 is a square. - Bruno Berselli, Apr 19 2013

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Barry Brent, On the constant terms of certain meromorphic modular forms for Hecke groups, arXiv:2212.12515 [math.NT], 2022.
Barry Brent, On the Constant Terms of Certain Laurent Series, Preprints (2023) 2023061164.

Cf. A001248, A049002, A065481.

Cf. A084920, A166010, A182200; A182174.

a049001 = subtract 2 . a001248  -- Reinhard Zumkeller, Jul 30 2015

A049001:=n->ithprime(n)^2-2; seq(A049001(k), k=1..50); # Wesley Ivan Hurt, Oct 11 2013

Table[Prime[n]^2-2, {n, 50}] (* Wesley Ivan Hurt, Oct 11 2013 *)

a(n) = prime(n)^2 - 2; \\ Amiram Eldar, Nov 07 2022

a(n) = A001248(n) - 2.
a(n) = A182200(n) + 1. - Wesley Ivan Hurt, Oct 11 2013
Product_{n>=1} (1 - 1/a(n)) = A065481. - Amiram Eldar, Nov 07 2022

A237413 Number of ways to write n = k + m with k > 0 and m > 0 such that p(k)^2 - 2, p(m)^2 - 2 and p(p(m))^2 - 2 are all prime, where p(j) denotes the j-th prime.

Original entry on oeis.org

0, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 3, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 5, 3, 1, 3, 3, 3, 3, 3, 1, 3, 1, 2, 2, 5, 2, 3, 3, 5, 2, 5, 7, 3, 3, 4, 5, 5, 5, 4, 4, 5, 2, 3, 4, 7, 5, 3, 4, 8, 6, 5, 4, 6, 5, 4, 2, 6, 5, 6, 5, 2, 6, 7

Offset: 1

Zhi-Wei Sun, Feb 07 2014

nonn

Conjecture: a(n) > 0 for all n > 1.
This conjecture was motivated by the "Super Twin Prime Conjecture".
See A237414 for primes q with q^2 - 2 and p(q)^2 - 2 both prime.

			a(7) = 1 since 7 = 6 + 1 with p(6)^2 - 2 = 13^2 - 2 = 167, p(1)^2 - 2 = 2^2 - 2 = 2 and p(p(1))^2 - 2 = p(2)^2 - 2 = 3^2 - 2 = 7 are all prime.
a(516) = 1 since 516 = 473 + 43 with p(473)^2 - 2 = 3359^2 - 2 = 11282879, p(43)^2 - 2 = 191^2 - 2 = 36479 and p(p(43))^2 - 2 = p(191)^2 - 2 = 1153^2 - 2 = 1329407 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Super Twin Prime Conjecture, a message to Number Theory List, Feb. 6, 2014.
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000040, A049002, A062326, A218829, A237348, A237367, A237414.

pq[k_]:=PrimeQ[Prime[k]^2-2]
a[n_]:=Sum[If[pq[k]&&pq[n-k]&&pq[Prime[n-k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,80}]

A137291 Numbers m such that prime(m)^2-2 is prime.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 14, 15, 18, 20, 24, 27, 28, 31, 32, 34, 40, 43, 47, 48, 51, 52, 55, 62, 65, 68, 72, 82, 86, 87, 91, 94, 99, 100, 104, 107, 111, 119, 123, 128, 129, 130, 132, 133, 134, 135, 139, 141, 150, 152, 170, 172, 177, 180, 182, 191, 200, 202, 209, 211

Offset: 1

Ctibor O. Zizka, Apr 05 2008

For m>=1, for these and only these numbers m, A242719(m) = prime(m)^2 + 1. Since A242719(m) >= prime(m)^2 + 1, then the equality is obtained on this and only this sequence. - Vladimir Shevelev, Sep 04 2014

			prime(24)*prime(24)-2 = 89*89-2 = 7919 is prime, so n=24 belongs to the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A062326, A242719, A242720, A246824, A010051, A049001, A103960, A210481.

a137291 n = a137291_list !! (n-1)
a137291_list = filter ((== 1) . a010051' . a049001) [1..]
-- Reinhard Zumkeller, Jul 30 2015

Select[Range[211],PrimeQ[Prime[#]^2-2]&] (* James C. McMahon, May 28 2025 *)

is(n,p=prime(n))=isprime(p^2-2) \\ Charles R Greathouse IV, Feb 17 2017

A103960(a(n)) - A210481(a(n)) = 1. - Reinhard Zumkeller, Jul 30 2015

a(n) = A049084(A049002(n)). - R. J. Mathar, Apr 09 2008

More terms from R. J. Mathar, Apr 09 2008

Offset corrected by Reinhard Zumkeller, Jul 30 2015

A261281 Least positive integer k with prime(k)^2-2 and prime(prime(k))^2-2 both prime such that prime(kn)^2-2 and prime(prime(kn))^2-2 are all prime.

Original entry on oeis.org

1, 1, 319, 134, 34, 62, 2, 536, 5215, 15, 3965, 2168, 34, 1, 1, 737, 2, 7075, 3699, 419, 132, 372, 14, 2, 34, 2, 52, 1, 668, 36561, 2, 48, 1239, 1, 401, 1613, 1646, 2472, 43, 31361, 134, 1103, 1, 5374, 6201, 466, 1, 1, 2118, 2, 1646, 1, 1343, 856, 28, 1868, 10324, 360, 2845, 6571, 65, 1, 419, 43, 1, 2, 2, 1, 889, 202

Offset: 1

Zhi-Wei Sun, Aug 14 2015

nonn

Conjecture: a(n) exists for any n > 0. In general, any positive rational number r can be written as m/n with m and n in the set {k>0: prime(k)^2-2 and prime(prime(k))^2-2 are both prime}.

This implies that the sequence A237414 has infinitely many terms.

			a(2) = 1 since prime(1)^2-2 = 2^2-2 = 2, prime(prime(1))^2-2 = prime(2)^2-2 = 3^2-2 = 7, prime(1*2)^2-2 = 3^2-2 = 7, and prime(prime(1*2))^2-2 = prime(3)^2-2 = 5^2-2 = 23 are all prime.
a(3) = 319 since prime(319)^2-2 = 2113^2-2 = 4464767, prime(prime(319))^2-2 = prime(2113)^2-2 = 18443^2-2 = 340144247, prime(319*3)^2-2 = 7547^2-2 = 56957207, and prime(prime(3*319))^2-2 = prime(7547)^2-2 = 76757^2-2 = 5891637047 are all prime.

Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Zhi-Wei Sun, Table of n, a(n) for n = 1..2000
Zhi-Wei Sun, Checking the conjecture for r = a/b with a,b = 1..300
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A049002, A062326, A237413, A237414, A259487.

f[n_]:=Prime[n]
q[n_]:=PrimeQ[f[n]^2-2]&&PrimeQ[f[f[n]]^2-2]
Do[k=0;Label[bb];k=k+1;If[q[k]&&q[k*n],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,70}]

a(n) = my(k=1); while (!isprime(prime(k)^2-2) || !isprime(prime(prime(k))^2-2) || !isprime(prime(k*n)^2-2) || !isprime(prime(prime(k*n))^2-2), k++); k; \\ Michel Marcus, Aug 14 2015

A238576 Number of odd primes p < 2n with prime(n(p-1)/2)^2 - 2 prime.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 1, 4, 3, 2, 2, 4, 2, 2, 2, 3, 2, 2, 4, 5, 2, 2, 1, 8, 2, 2, 3, 3, 2, 2, 4, 4, 5, 6, 2, 5, 4, 3, 3, 7, 2, 2, 8, 8, 5, 4, 6, 3, 3, 7, 6, 5, 3, 3, 9, 4, 8, 3, 5, 3, 1, 5, 6, 4, 6, 7, 7, 8, 6, 6, 2, 7, 1, 5, 9, 7, 5, 6, 5, 7

Offset: 1

Zhi-Wei Sun, Mar 01 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 7, 23, 61, 73.
(ii) For any integer n > 1, there is an odd prime p < 2*n with prime(n*(p+1)/2)^2 - 2 prime.
Clearly, either part of the conjecture implies that there are infinitely many primes of the form p^2 - 2 with p prime.

			a(2) = 1 since 2 and prime(2*(3-1)/2)^2 - 2 = 3^2 - 2 = 7 are both prime.
a(7) = 1 since 5 and prime(7*(5-1)/2)^2 - 2 = 43^2 - 2 = 1847 are both prime.
a(23) = 1 since 29 and prime(23*(29-1)/2)^2 - 2 = 2137^2 - 2 = 4566767 are both prime.
a(61) = 1 since 43 and prime(61*(43-1)/2)^2 - 2 = 10463^2 - 2 = 109474367 are both prime.
a(73) = 1 since 7 and prime(73*(7-1)/2)^2 - 2 = 1367^2 - 2 = 1868687 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.

Cf. A000040, A049002, A062326, A237413, A237578, A238573.

p[k_,n_]:=PrimeQ[Prime[(Prime[k]-1)/2*n]^2-2]
a[n_]:=Sum[If[p[k,n],1,0],{k,2,PrimePi[2n-1]}]
Table[a[n],{n,1,80}]

A115093 Primes of the form p*q-2, where p and q are distinct primes.

Original entry on oeis.org

13, 19, 31, 37, 53, 67, 83, 89, 109, 113, 127, 131, 139, 157, 181, 199, 211, 233, 251, 257, 263, 293, 307, 317, 337, 353, 379, 389, 401, 409, 443, 449, 467, 479, 487, 491, 499, 503, 509, 541, 557, 563, 571, 577, 587, 631, 647, 653, 677, 683, 701, 719, 743

Offset: 1

T. D. Noe, Mar 01 2006

nonn

This sequence is a subset of A063637, which is the union of this sequence and A049002.

Cf. A049002 (primes of the form p^2-2, where p is prime), A063637 (primes of the form p*q-2 where p and q are primes).

SemiPrimeQ[n_] := (n>1) && (2==Plus@@(Transpose[FactorInteger[n]][[2]])); Select[Prime[Range[150]], SemiPrimeQ[ #+2] && !IntegerQ[Sqrt[ #+2]]&]

A238701 Number of primes p < n with q = floor((n-p)/4) and q^2 - 2 both prime.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 3, 3, 4, 4, 4, 3, 3, 3, 3, 4, 4, 4, 6, 5, 5, 5, 3, 4, 6, 6, 7, 6, 4, 4, 4, 4, 5, 5, 5, 5, 4, 4, 4, 4, 3, 3, 4, 4, 6, 6, 4, 5, 5, 5, 7, 6, 6, 6, 5, 5, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 5, 5, 5, 5, 3, 4, 5, 5

Offset: 1

Zhi-Wei Sun, Mar 03 2014

nonn

Conjecture: Let m > 0 and n > 2*m + 1 be integers. If m = 1 and 2 | n, or m = 3 and n is not congruent to 1 modulo 6, or m = 2, 4, 5, ..., then there is a prime p < n with q = floor((n-p)/m) and q^2 - 2 both prime.

In the case m = 1, this is a refinement of Goldbach's conjecture. In the case m = 2, this is stronger than Lemoine's conjecture (cf. A046927). The conjecture for m > 2 seems completely new. We view the conjecture as a natural extension of Goldbach's conjecture.

			a(11) = 2 since 2, floor((11-2)/4)= 2 and 2^2 - 2 are all prime, and 3, floor((11-3)/4) = 2 and 2^2 - 2 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.

Cf. A000040, A045917, A046927, A049002, A062326, A230502, A237413, A237705.

PQ[n_]:=PrimeQ[n]&&PrimeQ[n^2-2]
p[n_,k_]:=PQ[Floor[(n-Prime[k])/4]]
a[n_]:=Sum[If[p[n,k],1,0],{k,1,PrimePi[n-1]}]
Table[a[n],{n,1,80}]

A261354 Primes p such that prime(p)^2 - 2 = prime(q) for some prime q.

Original entry on oeis.org

31, 191, 541, 809, 1153, 1301, 2221, 3037, 3847, 4049, 4159, 5441, 8243, 10177, 12277, 13681, 14783, 15619, 17903, 19463, 20897, 22697, 24517, 25163, 25847, 25849, 26633, 26647, 27329, 27407, 28051, 32653, 35059, 35747, 36341, 36527, 37369, 37811, 38609, 40949, 42737, 46679, 51061, 51607, 54443, 54679, 56113, 57637, 60887, 61493

Offset: 1

Zhi-Wei Sun, Aug 15 2015

nonn

Conjecture: The sequence has infinitely many terms. In general, for any integers a,b,c with a>0 and gcd(a,b,c)=1, if b^2-4*a*c is not a square, a+b+c is odd, and gcd(b,a+c) is not divisible by 3, then there are infinitely many prime pairs {p,q} such that a*prime(p)^2+b*prime(p)+c = prime(q).

			a(1) = 31 since 31 is a prime, and prime(31)^2-2 = 127^2-2 = 16127 = prime(1877) with 1877 prime.

Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A049002, A062326, A237413, A260120, A261281, A261352, A261361.

PQ[n_]:=PrimeQ[n]&&PrimeQ[PrimePi[n]]
f[k_]:=Prime[Prime[k]]^2-2
n=0;Do[If[PQ[f[k]],n=n+1;Print[n," ",Prime[k]]],{k,1,6200}]

cp's OEIS Frontend

A062326 Primes p such that p^2 - 2 is also prime.

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A063637 Primes p such that p+2 is a semiprime.

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A049001 a(n) = prime(n)^2 - 2.

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A237413 Number of ways to write n = k + m with k > 0 and m > 0 such that p(k)^2 - 2, p(m)^2 - 2 and p(p(m))^2 - 2 are all prime, where p(j) denotes the j-th prime.

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A137291 Numbers m such that prime(m)^2-2 is prime.

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A261281 Least positive integer k with prime(k)^2-2 and prime(prime(k))^2-2 both prime such that prime(k*n)^2-2 and prime(prime(k*n))^2-2 are all prime.

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A238576 Number of odd primes p < 2*n with prime(n*(p-1)/2)^2 - 2 prime.

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A261281 Least positive integer k with prime(k)^2-2 and prime(prime(k))^2-2 both prime such that prime(kn)^2-2 and prime(prime(kn))^2-2 are all prime.

A238576 Number of odd primes p < 2n with prime(n(p-1)/2)^2 - 2 prime.