A118915 -id:A118915 - cp's OEIS frontend

A109953 Primes p such that p^2+2 is a semiprime.

2, 7, 11, 17, 29, 37, 43, 53, 73, 79, 83, 97, 137, 191, 233, 251, 263, 269, 271, 277, 281, 359, 379, 389, 433, 461, 479, 521, 541, 577, 601, 631, 647, 677, 691, 719, 739, 827, 829, 863, 881, 929, 947, 983, 997, 1033, 1063, 1087, 1109, 1187, 1223

Offset: 1

Zak Seidov, Jul 06 2005

nonn

Cf. A048161 Primes p such that p^2+1 is a semiprime.

Primes p such that (p^2+2)/3 is prime. For all primes q>3, we have q=6k+-1 for some k, which makes it easy to show that 3 divides q^2+2. Hence if q^2+2 is a semiprime then (q^2+2)/3 must be prime. - T. D. Noe, May 05 2006

			a(2) = 7 is o.k. because 7^2+2=51=3*17 (semiprime), and 17 = A289135(2).

Harvey P. Dale, Table of n, a(n) for n = 1..2500

Cf. A048161, A289135.

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

A109953=Select[Prime[Range[200]], Plus@@Last/@FactorInteger[ #^2+2]==2&]
Select[Prime[Range[200]],PrimeOmega[#^2+2]==2&] (* Harvey P. Dale, Nov 19 2011 *)

a(n) = sqrt(3*A289135(n) - 2). See the T. D. Noe comment above. - Wolfdieter Lang, Jul 19 2017

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

A247478 Primes p such that (p^4 + 5)/6 is prime.

Original entry on oeis.org

7, 11, 17, 29, 53, 71, 101, 109, 127, 179, 227, 241, 281, 307, 349, 487, 587, 647, 683, 727, 829, 1009, 1061, 1109, 1289, 1487, 1511, 1523, 1567, 1621, 1627, 1709, 1847, 1987, 2017, 2027, 2087, 2099, 2297, 2311, 2393, 2437, 2447, 2521, 2531, 2617, 2729, 2887, 2909, 2969, 3167, 3221, 3301, 3319, 3329, 3347, 3413, 3527

Offset: 1

Zak Seidov, Jan 19 2015

nonn

(p^4+5)/6 is an integer for all primes p>3, because then p == (1 or 5) (mod 6) as in A039704, therefore p^2 == 1 (mod 6) and finally p^4 == 1 (mod 6).

			(7^4+5)/6 = 401 prime, (11^4+5)/6 = 2441 prime.

Cf. A118915.

[p: p in PrimesInInterval(3, 4000) | IsPrime((p^4+5) div 6)]; //  Vincenzo Librandi, Jan 21 2015

Select[Prime[Range[10^3]], PrimeQ[(#^4 + 5) / 6] &] (* Vincenzo Librandi, Jan 21 2015 *)

lista(nn) = {forprime(p=4, nn, if (isprime((p^4 + 5)/6), print1(p, ", ")););} \\ Michel Marcus, Jan 20 2015

A253940 Primes p such that (p^2 + 5)/6, (p^4 + 5)/6, and (p^8 + 5)/6 are prime.

Original entry on oeis.org

39367, 52163, 67103, 79631, 100981, 280547, 318457, 530711, 605123, 815401, 833923, 834947, 928871, 1313857, 1734067, 1750069, 1800973, 2163979, 2427137, 2598119, 2611027, 2754991, 2764187, 2836259, 3040757, 3101309, 3118697, 3465953, 3646693, 4014809

Offset: 1

Zak Seidov, Jan 20 2015

nonn

R. J. Mathar, Table of n, a(n) for n = 1..108

Subsequence of A253925. Cf. A118915, A247478, A253939.

[p: p in PrimesUpTo(10^7) | IsPrime((p^2+5) div 6) and IsPrime((p^4+5) div 6) and IsPrime((p^8+5) div 6)]; // Vincenzo Librandi, Jan 21 2015

Select[Prime[Range[10^6]], PrimeQ[(#^2 + 5) / 6] &&PrimeQ[(#^4 + 5) / 6] &&PrimeQ[(#^8 + 5) / 6] &] (* Vincenzo Librandi, Jan 21 2015 *)
Select[Prime[Range[300000]],AllTrue[({#^2,#^4,#^8}+5)/6,PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 15 2021 *)

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

A253925 Primes p such that both (p^2 + 5)/6 and (p^4 + 5)/6 are prime.

Original entry on oeis.org

127, 1009, 1709, 2087, 2393, 2969, 3221, 3347, 7309, 7757, 7883, 10529, 11411, 12923, 17569, 18269, 21799, 23311, 23633, 24877, 25703, 26839, 27091, 29429, 35461, 35603, 38431, 39367, 39761, 41887, 42967, 43037, 45361, 45989, 47699, 52163, 59093, 63629, 65323, 67103, 68041, 69481, 70937, 74843, 77813, 77867

Offset: 1

Zak Seidov, Jan 19 2015

nonn

Intersection of A118915 and A247478.

			a(1)=127=A118915(9)=A247478(9). a(2)=1009=A118915(45)=A247478(22).

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

Cf. A118915, A247478.

forprime(p=5,1e6,if(isprime((p^2+5)/6) && isprime((p^4+5)/6), print1(p", "))) \\ Charles R Greathouse IV, Jan 19 2015

A253941 Primes p such that (p^2 + 5)/6, (p^4 + 5)/6, (p^6 + 5)/6, (p^8 + 5)/6 and (p^10 + 5)/6 are all prime.

Original entry on oeis.org

184279409, 619338131, 913749803, 1057351301, 1507289869, 1600204213, 2845213937, 4725908767, 4760956439, 5374709801, 5518707641, 8724256757, 9044067313, 9387396269, 10992352517, 11937043567, 13493126359, 13593105793, 17891702891, 17897035213, 17954907767, 19690938161, 20227580927, 20922685813, 21313027583, 21717176851

Offset: 1

Zak Seidov, Jan 20 2015

nonn

The sequence contains all terms up to 10^10. There are no terms as yet for which (p^12 + 5)/6 is also prime.

No terms < 10^11 with (p^12 + 5)/6 prime. - Chai Wah Wu, Jan 27 2015

Chai Wah Wu, Table of n, a(n) for n = 1..67

Subsequence of A253976.

Cf. A118915, A247478, A253925, A253940.

lista(nn) = forprime(p=5, nn, if(ispseudoprime((p^2 + 5)/6) && ispseudoprime((p^4 + 5)/6) && ispseudoprime((p^6 + 5)/6) && ispseudoprime((p^8 + 5)/6) && ispseudoprime((p^10 + 5)/6), print1(p, ", "))); \\ Jinyuan Wang, Mar 01 2020

from gmpy2 import is_prime, t_divmod
A253941_list = []
for p in range(1,10**6,2):
    if is_prime(p):
        p2, x = p**2, 1
        for i in range(5):
            x *= p2
            q, r = t_divmod(x+5,6)
            if r or not is_prime(q):
                break
        else:
            A253941_list.append(p) # Chai Wah Wu, Jan 22 2015

a(15)-a(26) from Chai Wah Wu, Jan 22 2015

A253976 Primes p such that (p^2 + 5)/6, (p^4 + 5)/6, (p^6 + 5)/6 and (p^8 + 5)/6 are prime.

Original entry on oeis.org

67103, 7524593, 9938069, 10125793, 13042637, 55741139, 55792241, 58429099, 77618323, 92713879, 94554613, 96242761, 103774049, 119753549, 141725501, 142915193, 164899799, 165227399, 173202247, 174728233, 178411771, 184279409, 184356703, 186622003, 195863347, 200406977, 239488649

Offset: 1

Zak Seidov, Jan 21 2015

nonn

Zak Seidov, Table of n, a(n) for n = 1..503 (all terms up to 10^10)

Subsequence of A253939. Cf. A118915, A247478, A253939, A253940, A253941.

Select[Prime[Range[132*10^5]],AllTrue[(#^Range[2,8,2]+5)/6,PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 11 2018 *)

forprime(p=1,10^7,k=0;for(i=1,4,P=(p^(2*i)+5)/6;if(P\1==P,if(ispseudoprime(P),k++);if(!ispseudoprime(P),k=0;break));if(P\1!=P,k=0;break));if(k,print1(p,", "))) \\ Derek Orr, Jan 21 2015

cp's OEIS Frontend

A109953 Primes p such that p^2+2 is a semiprime.

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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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A247478 Primes p such that (p^4 + 5)/6 is prime.

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Magma

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A253940 Primes p such that (p^2 + 5)/6, (p^4 + 5)/6, and (p^8 + 5)/6 are prime.

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Magma

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

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Mathematica

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

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Maple

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A253925 Primes p such that both (p^2 + 5)/6 and (p^4 + 5)/6 are prime.

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A253941 Primes p such that (p^2 + 5)/6, (p^4 + 5)/6, (p^6 + 5)/6, (p^8 + 5)/6 and (p^10 + 5)/6 are all prime.

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