A118940 -id:A118940 - cp's OEIS frontend

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

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]&]

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]&]

A169583 n-th prime*8-7 is the square of a prime.

Original entry on oeis.org

1, 4, 12, 19, 47, 59, 115, 167, 217, 251, 306, 348, 414, 618, 630, 662, 809, 1077, 1138, 1218, 1670, 1876, 2272, 2680, 2869, 3402, 3633, 4242, 4353, 4661, 5255, 6463, 6596, 6986, 8543, 8870, 8992, 9340, 9444, 10265, 11544, 11921, 12449, 13887, 14031

Offset: 1

Juri-Stepan Gerasimov, Mar 01 2010

nonn

			1 is in the sequence because 1st prime*8-7=(2nd prime)^2;
4 is in the sequence because 4th prime*8-7=(4th prime)^2;
12 is in the sequence because 12th prime*8-7=(7th prime)^2.

Harvey P. Dale, Table of n, a(n) for n = 1..350

Cf. A000040, A001248, A055469, A118940.

Select[Range[15000],PrimeQ[Sqrt[8Prime[#]-7]]&] (* Harvey P. Dale, Nov 10 2017 *)

isok(n) = issquare(sq=prime(n)*8-7) && isprime(sqrtint(sq)); \\ Michel Marcus, Apr 15 2014

Corrected (1077 inserted) and extended by R. J. Mathar, Jun 04 2010

A299691 Smallest prime p that remains prime through exactly n iterations of the function f(x) = (x^2 + 7)/8.

Original entry on oeis.org

2, 3, 89, 263, 386777, 149953319

Offset: 0

Jon E. Schoenfield, Feb 16 2018

Since f(7) = (7^2 + 7)/8 = 7, p=7 remains prime through infinitely many iterations of the function.

			f(2) = (2^2 + 7)/8 = 11/8 (not a prime), and 2 is the smallest prime, so a(0) = 2.
f(3) = (3^2 + 7)/8 = 16/8 = 2 (prime), but (2^2 + 7)/8 = 11/8 (not a prime), so p remains prime through exactly one iteration, and p=3 is the smallest prime for which this is the case, so a(1) = 3.
f(89) = (89^2 + 7)/8 = 991 (prime), and f(991) = (991^2 + 7)/8 = 122761 (prime), but f(122761) = (122761^2 + 7)/8 = 1883782891 = 211 * 8927881 (not a prime), so p remains prime through exactly two iterations, and p=89 is the smallest prime for which this is the case, so a(2) = 89.

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

Block[{lim = 10^2, s}, s = Array[Length@ NestWhileList[(#^2 + 7)/8 &, Prime@ #, PrimeQ, 1, lim, -1] /. lim -> 0 &, 10^6]; Array[Prime@ FirstPosition[s, #][[1]] &, Max@ s]] (* Michael De Vlieger, Feb 18 2018 *)

isprimeq(q) = {if (denominator(q) != 1, return (0)); isprime(q);}
isok(p, n) = {for (k=1, n, q = (p^2 + 7)/8; if (! isprimeq(q), return (0)); p = q;); q = (p^2 + 7)/8; return (! isprimeq(q));}
a(n) = {forprime(p=2, , if (isok(p, n), return (p)););} \\ Michel Marcus, Feb 26 2018

cp's OEIS Frontend

A118939 Primes p such that (p^2+3)/4 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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A169583 n-th prime*8-7 is the square of a prime.

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A299691 Smallest prime p that remains prime through exactly n iterations of the function f(x) = (x^2 + 7)/8.

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Mathematica

PARI