A109953 -id:A109953 - cp's OEIS frontend

A118915 Primes p such that (p^2 + 5)/6 is prime.

5, 13, 19, 23, 37, 41, 89, 113, 127, 131, 139, 149, 167, 197, 229, 233, 239, 251, 271, 359, 373, 401, 433, 449, 463, 503, 523, 541, 607, 631, 643, 653, 701, 719, 743, 769, 811, 827, 877, 881, 887, 919, 967, 971, 1009, 1013, 1021, 1093, 1097, 1283, 1301

Offset: 1

T. D. Noe, May 05 2006

nonn

For all primes q>3, we have q=6k+-1 for some k, which makes it easy to show that 6 divides q^2+5.

(n^2+5)/6 is an integer for all primes except 2 and 3. - Michael B. Porter, Apr 14 2010

Cf. A109953 (primes p such that (p^2+1)/3 is prime), A118918 (primes p such that (p^2+11)/12 is prime).

Select[Prime[Range[215]],PrimeQ[(#^2+5)/6]&] (* James C. McMahon, Sep 13 2024 *)

isA118915(n)=if(n^2%6==1,isprime(n)&&isprime((n^2+5)/6),0) \\ Michael B. Porter, Apr 14 2010

A118918 Primes p such that (p^2+11)/12 is prime.

Original entry on oeis.org

5, 7, 11, 19, 29, 61, 71, 79, 89, 109, 151, 179, 181, 191, 199, 271, 281, 349, 379, 389, 421, 439, 479, 521, 541, 569, 631, 659, 691, 809, 821, 829, 839, 919, 971, 1019, 1051, 1061, 1069, 1091, 1289, 1439, 1511, 1621, 1699, 1709, 1789, 1811, 1871, 2069, 2141

Offset: 1

T. D. Noe, May 05 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+11.

Cf. A109953 (primes p such that (p^2+1)/3 is prime), A118915 (primes p such that (p^2+11)/12 is prime).

Select[Prime[Range[400]],PrimeQ[(#^2+11)/12]&]

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

Original entry on oeis.org

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]&]

A241120 Primes p such that (p^3 + 2)/3 is prime.

Original entry on oeis.org

13, 19, 31, 193, 211, 223, 229, 271, 331, 571, 619, 691, 739, 751, 853, 991, 1009, 1039, 1051, 1231, 1303, 1321, 1549, 1741, 1789, 1831, 1993, 1999, 2029, 2089, 2113, 2143, 2203, 2311, 2521, 2551, 2683, 2749, 2851, 3121, 3259, 3331, 3571, 3631, 3823, 3853, 4093

Offset: 1

K. D. Bajpai, Apr 16 2014

nonn

			13 is prime and appears in the sequence because (13^3 + 2)/3 = 733 which is a prime.
31 is prime and appears in the sequence because (31^3 + 2)/3 = 9931 which is a prime.

K. D. Bajpai, Table of n, a(n) for n = 1..8879

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).
Cf. A241101 (primes p: (p^3-4)/3 is prime).

KD:= proc() local a,b;a:=ithprime(n); b:=(a^3+2)/3; if b=floor(b) and isprime(b) then RETURN (a); fi; end: seq(KD(), n=1..1000);

Select[Prime[Range[500]], PrimeQ[(#^3 + 2)/3] &]
n = 0; Do[If[PrimeQ[(Prime[k]^3 + 2)/3], n = n + 1; Print[n, " ", Prime[k]]], {k, 1, 200000}]          (* b-file *)

s=[]; forprime(p=2, 8000, if((p^3+2)%3==0 && isprime((p^3+2)/3), s=concat(s, p))); s \\ Colin Barker, Apr 16 2014

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]&]

A241101 Primes p such that (p^3 - 4)/3 is prime.

Original entry on oeis.org

7, 31, 37, 43, 61, 73, 97, 103, 157, 163, 211, 277, 331, 337, 457, 487, 613, 661, 733, 751, 811, 883, 991, 1021, 1093, 1297, 1321, 1483, 1693, 1741, 1873, 2083, 2113, 2143, 2203, 2221, 2287, 2347, 2437, 2473, 2707, 2917, 3001, 3067, 3187, 3307, 3331, 3343, 3541

Offset: 1

K. D. Bajpai, Apr 16 2014

nonn

			7 is prime and appears in the sequence because (7^3 - 4)/3 = 113 which is a prime.
31 is prime and appears in the sequence because (31^3 - 4)/3 = 9929 which is a prime.

K. D. Bajpai, Table of n, a(n) for n = 1..8906

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*)

s=[]; forprime(p=2, 4000, if((p^3-4)%3==0 && isprime((p^3-4)/3), s=concat(s, p))); s \\ Colin Barker, Apr 16 2014

A289135 Prime numbers p such that 3*p - 2 is the square of a prime number.

Original entry on oeis.org

2, 17, 41, 97, 281, 457, 617, 937, 1777, 2081, 2297, 3137, 6257, 12161, 18097, 21001, 23057, 24121, 24481, 25577, 26321, 42961, 47881, 50441, 62497, 70841, 76481, 90481, 97561, 110977, 120401, 132721, 139537, 152777, 159161, 172321, 182041

Offset: 1

Dimitris Valianatos, Jun 25 2017

nonn

Terms > 2 are congruent to either 1 or 17 mod 40. - Davide Rotondo, Feb 06 2024

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A109953.

Select[Prime@ Range@ 20000, PrimeQ@ Sqrt[3 # - 2] &] (* Michael De Vlieger, Jun 26 2017 *)

forprime(n=2,10000,if(isprimepower(3*n-2)==2,print1(n", ")))

list(lim)=my(v=List([2]),p); forprime(q=7,sqrtint(lim\1*3-2), if(isprime(p=(q^2+2)/3), listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Jul 16 2017

a(n) = (A109953(n)^2 + 2) / 3.

A235705 Primes p such that (p^3 + 6)/5 is prime.

Original entry on oeis.org

19, 59, 269, 349, 409, 419, 479, 769, 929, 1109, 1319, 1399, 1979, 2609, 3659, 4079, 4919, 5309, 5449, 5879, 6079, 6299, 6949, 7069, 7129, 7229, 7699, 7829, 8069, 8329, 8599, 9679, 10729, 11969, 12809, 13109, 13229, 13859, 14159, 14419, 14629, 14929, 15259

Offset: 1

K. D. Bajpai, Apr 20 2014

nonn

All the terms in the sequence are congruent to 1 or 3 mod 4.

			a(1) = 19 is prime: (19^3 + 6)/ 5 = 1373 which is also prime.
a(2) = 59 is prime: (59^3 + 6)/ 5 = 41077 which is also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..2640

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).
Cf. A241101 (primes p: (p^3-4)/3 is prime).
Cf. A241120 (primes p: (p^3+2)/3 is prime).

KD:= proc() local a,b; a:=ithprime(n); b:=(a^3+6)/5; if b=floor(b) and isprime(b) then RETURN (a); fi; end: seq(KD(), n=1..5000);

Select[Prime[Range[5000]], PrimeQ[(#^3 + 6)/5] &]
n = 0; Do[If[PrimeQ[(Prime[k]^3 + 6)/5], n = n + 1; Print[n, " ", Prime[k]]], {k, 1, 200000}] (*b-file*)

s=[]; forprime(p=2, 20000, if((p^3+6)%5==0 && isprime((p^3+6)/5), s=concat(s, p))); s \\ Colin Barker, Apr 21 2014

cp's OEIS Frontend

A118915 Primes p such that (p^2 + 5)/6 is prime.

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A118918 Primes p such that (p^2+11)/12 is prime.

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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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A241120 Primes p such that (p^3 + 2)/3 is 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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A241101 Primes p such that (p^3 - 4)/3 is prime.

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A289135 Prime numbers p such that 3*p - 2 is the square of a prime number.

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