A062718 -id:A062718 - cp's OEIS frontend

A118940 Primes p such that (p^2+7)/8 is prime.

3, 7, 17, 23, 41, 47, 71, 89, 103, 113, 127, 137, 151, 191, 193, 199, 223, 263, 271, 281, 337, 359, 401, 439, 457, 503, 521, 569, 577, 599, 641, 719, 727, 751, 839, 857, 863, 881, 887, 929, 991, 1009, 1033, 1097, 1103, 1151, 1193, 1217, 1231, 1279, 1297, 1303

Offset: 1

T. D. Noe, May 06 2006

nonn

For all primes q>2, we have q=4k+-1 for some k, which makes it easy to show that 8 divides q^2+7.

Similar sequences, with p and (p^2+a)/b both prime, are A048161, A062324, A062326, A062718, A109953, A110589, A118915, A118918, A118939, A118941 and A118942.

Select[Prime[Range[200]],PrimeQ[(#^2+7)/8]&]

lista(nn) = {forprime(p=2, nn, iferr(if (isprime(q=(p^2+7)/8), print1(q, ", ")), E, ););} \\ Michel Marcus, Feb 18 2018

A118939 Primes p such that (p^2+3)/4 is prime.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 29, 31, 41, 43, 67, 83, 101, 109, 139, 151, 157, 179, 181, 199, 211, 223, 239, 263, 277, 283, 307, 311, 337, 347, 353, 379, 389, 419, 431, 463, 491, 557, 577, 587, 619, 659, 673, 739, 757, 797, 809, 811, 829, 853, 907, 911, 953, 991, 1051

Offset: 1

T. D. Noe, May 06 2006

nonn

For all primes q>2, we have q=4k+-1 for some k, which makes it easy to show that 4 divides q^2+3. Similar sequences, with p and (p^2+a)/b both prime, are A048161, A062324, A062326, A062718, A109953, A110589, A118915, A118918, A118940, A118941 and A118942.

Select[Prime[Range[200]],PrimeQ[(#^2+3)/4]&]

A137270 Primes p such that p^2 - 6 is also prime.

Original entry on oeis.org

3, 5, 7, 13, 17, 23, 47, 53, 67, 73, 83, 97, 107, 113, 167, 193, 197, 263, 293, 317, 367, 373, 383, 457, 463, 467, 487, 503, 557, 593, 607, 643, 647, 673, 677, 683, 773, 787, 797, 823, 827, 857, 877, 887, 947, 1033, 1063, 1087, 1103, 1187, 1193, 1223, 1303

Offset: 1

Ben de la Rosa and Johan Meyer (meyerjh.sci(AT)ufa.ac.za), Mar 13 2008

Each of the primes p = 2,3,5,7,13 has the property that the quadratic polynomial phi(x) = x^2 + x - p^2 takes on only prime values for x = 1,2,...,2p-2; each case giving exactly one repetition, in phi(p-1) = -p and phi(p) = p.

The only common term in A062718 and A137270 is 5. - Zak Seidov, Jun 16 2015

			The (2 x 7 - 2) -1 = 11 primes given by the polynomial x^2 + x - 7^2 for x = 1, 2, ..., 2 x 7 - 2 are -47, -43, -37, -29, -19, -7, 7, 23, 41, 61, 83, 107.

F. G. Frobenius, Uber quadratische Formen, die viele Primzahlen darstellen, Sitzungsber. d. Konigl. Acad. d. Wiss. zu Berlin, 1912, 966 - 980.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A062326, A062718.

[p: p in PrimesUpTo(1350) | IsPrime(p^2-6)]; // Vincenzo Librandi, Apr 14 2013

isA028879 := proc(n) isprime(n^2-6) ; end: isA137270 := proc(n) isprime(n) and isA028879(n) ; end: for i from 1 to 300 do if isA137270(ithprime(i)) then printf("%d, ",ithprime(i)) ; fi ; od: # R. J. Mathar, Mar 16 2008

Select[Prime[Range[2,300]],PrimeQ[#^2-6]&] (* Harvey P. Dale, Jul 24 2012 *)

A000040 INTERSECT A028879. - R. J. Mathar, Mar 16 2008

Corrected and extended by R. J. Mathar, Mar 16 2008

A118941 Primes p such that (p^2-5)/4 is prime.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 23, 31, 41, 43, 53, 61, 71, 79, 83, 89, 97, 101, 107, 109, 113, 131, 137, 167, 173, 179, 193, 229, 241, 251, 263, 269, 277, 281, 283, 307, 311, 317, 349, 353, 373, 383, 419, 431, 439, 461, 463, 467, 563, 571, 577, 593, 607, 613, 619, 647

Offset: 1

T. D. Noe, May 06 2006

nonn

For all primes q>2, we have q=4k+-1 for some k, which makes it easy to show that 4 divides q^2-5. Similar sequences, with p and (p^2+a)/b both prime, are A048161, A062324, A062326, A062718, A109953, A110589, A118915, A118918, A118939, A118940 and A118942.

Select[Prime[Range[200]],PrimeQ[(#^2-5)/4]&]

A118942 Primes p such that (p^2-13)/12 is prime.

Original entry on oeis.org

7, 13, 17, 19, 23, 31, 37, 41, 53, 67, 71, 73, 89, 103, 107, 113, 131, 139, 157, 163, 181, 199, 211, 233, 239, 257, 269, 283, 307, 311, 337, 359, 373, 379, 401, 419, 463, 487, 491, 499, 509, 521, 577, 593, 607, 617, 631, 647, 653, 683, 701, 733, 761, 769, 787

Offset: 1

T. D. Noe, May 06 2006

nonn

For all primes q>3, we have q=6k+-1 for some k, which makes it easy to show that 12 divides q^2-13. Similar sequences, with p and (p^2+a)/b both prime, are A048161, A062324, A062326, A062718, A109953, A110589, A118915, A118918, A118939, A118940 and A118941.

Select[Prime[Range[200]],PrimeQ[(#^2-13)/12]&]

A173627 Primes p such that p^2 + 6, p^2 + 12 and p^2 + 18 are all prime.

Original entry on oeis.org

5, 19, 61, 971, 1451, 2711, 3061, 3449, 6011, 15139, 15241, 21821, 27851, 39839, 51749, 62459, 75679, 76081, 82591, 97001, 121039, 121441, 122299, 135581, 161569, 162671, 196681, 196831, 200881, 214741, 217271, 222931, 242069, 243119, 254161

Offset: 1

Zak Seidov, Nov 09 2010

nonn

For p > 2, p^2 + 24 is composite (divisible by 5). - Zak Seidov, Sep 07 2018

Zak Seidov, Table of n, a(n) for n = 1..1800

Cf. A062718 (p and p^2 + 6 are both prime).

[p: p in PrimesUpTo(600000)|IsPrime(p^2+6) and IsPrime(p^2+12) and IsPrime(p^2+18)] // Vincenzo Librandi, Dec 13 2010

okQ[n_]:=Module[{p2=n^2},And@@PrimeQ[{p2+6,p2+12,p2+18}]]; Select[Prime[Range[30000]],okQ] (* Harvey P. Dale, Dec 18 2010 *)

isok(p) = isprime(p) && isprime(p^2+6) && isprime(p^2+12) && isprime(p^2+18); \\ Michel Marcus, Sep 08 2018

A245048 Primes p such that p^2 + 28 is prime.

Original entry on oeis.org

3, 5, 11, 13, 17, 19, 23, 41, 43, 47, 53, 67, 79, 83, 89, 97, 109, 131, 137, 149, 157, 163, 167, 179, 181, 193, 211, 223, 239, 241, 251, 263, 277, 281, 311, 317, 331, 379, 397, 401, 409, 421, 431, 439, 443, 449, 457, 467, 479, 541, 569, 599, 643, 647, 673

Offset: 1

Chai Wah Wu, Jul 10 2014

nonn

7 of the first 8 odd primes are in this list.

			3 is in the sequence because 3^2 + 28 = 37, which is also prime.
5 is in the sequence because 5^2 + 28 = 53, which is also prime.
7 is not in the sequence because 7^2 + 28 = 77 = 7 * 11.

Chai Wah Wu, Table of n, a(n) for n = 1..2000

Cf. A062324 (p^2+4), A062718(p^2+6), A243367(p^2+10).

A245048:=n->`if`(isprime(n) and isprime(n^2+28), n, NULL): seq(A245048(n), n=1..10^3); # Wesley Ivan Hurt, Jul 24 2014

Select[Prime[Range[200]], PrimeQ[#^2 + 28] &] (* Alonso del Arte, Jul 12 2014 *)

import sympy
[sympy.prime(n) for n in range(1,10**6) if sympy.ntheory.isprime(sympy.prime(n)**2+28)]

A258992 Primes p such that p^2 - 8 is also prime.

Original entry on oeis.org

5, 7, 11, 17, 19, 23, 31, 37, 53, 67, 101, 103, 149, 163, 173, 191, 227, 229, 241, 257, 269, 271, 313, 347, 353, 359, 367, 373, 383, 431, 467, 479, 487, 523, 541, 563, 577, 599, 613, 619, 647, 653, 661, 733, 761, 773, 823, 829, 859, 863, 919, 941, 1061, 1087

Offset: 1

Zak Seidov, Jun 16 2015

nonn

The first appearances of 2..6 consecutive primes in the sequence are: {31,37}, {5, 7, 11}, {353, 359, 367, 373}, {1293199, 1293203, 1293233, 1293239, 1293247}, {3982031, 3982037, 3982057, 3982067, 3982073, 3982079}.

Initial terms of the sets of exactly 6 consecutive primes: {3982031, 5495989, 33057589, 255414437, 495180067, 558985507}.

			From _K. D. Bajpai_, Jun 18 2015: (Start)
a(3) = 11: both 11 and 11^2 - 8 = 113 are prime.
a(4) = 17: both 17 and 17^2 - 8 = 281 are prime.
(End)

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A062718, A137270.

[p: p in PrimesUpTo(5000) | IsPrime(p^2-8)];  // K. D. Bajpai, Jun 18 2015

Select[Prime[Range[5000]], PrimeQ[#^2-8]&]  (* K. D. Bajpai, Jun 18 2015 *)

lista(nn) = forprime(p=2, nn, if (isprime(p^2-8), print1(p, ", "))); \\ Michel Marcus, Jun 16 2015

cp's OEIS Frontend

A118940 Primes p such that (p^2+7)/8 is prime.

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A118939 Primes p such that (p^2+3)/4 is prime.

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A137270 Primes p such that p^2 - 6 is also prime.

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A118941 Primes p such that (p^2-5)/4 is prime.

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A118942 Primes p such that (p^2-13)/12 is prime.

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A173627 Primes p such that p^2 + 6, p^2 + 12 and p^2 + 18 are all prime.

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A245048 Primes p such that p^2 + 28 is prime.

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A258992 Primes p such that p^2 - 8 is also prime.

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Magma

Mathematica