A062324 -id:A062324 - cp's OEIS frontend

A165218 Primes q of the form q=p^2+4 (p=prime) such that r=q^2+4 is also prime.

13, 293, 10613, 18773, 76733, 97973, 458333, 552053, 1247693, 2647133, 4012013, 4592453, 11607653, 13520333, 20097293, 25877573, 34845413, 51509333, 53772893, 65399573, 65496653, 66373613, 72880373, 73496333, 86359853, 89737733

Offset: 1

Zak Seidov, Sep 08 2009

nonn

Intersection of A062324 and A045637. Except of the first term, 13, all terms == 5 (mod 6) == 5 (mod 12) == 5 (mod 24) == 23 (mod 30)== 53 (mod 120). Values of primes p in A116886.

			Prime q=13=p^2+4 (p=3) and r=q^2+4=13^2+4=173 (prime).
Prime q=293=p^2+4 (p=17) and r=q^2+4=293^2+4=85853 (prime).

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A045637, A062324, A116886.

Reap[For[p = 2, p < 10^4, p = NextPrime[p], If[PrimeQ[q = p^2+4] && PrimeQ[q^2+4], Print[q]; Sow[q]]]][[2, 1]] (* Jean-François Alcover, Nov 07 2013 *)
Select[Prime[Range[2000]]^2+4,AllTrue[{#,#^2+4},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 13 2018 *)

a(n) = (A116886(n))^2 + 4.

A116888 Primes p that remain prime through 4 iterations of the function f(p) = p^2 + 4.

Original entry on oeis.org

306167, 48639197, 64695713, 68252687, 87788237, 87813293, 160486967, 255974437, 283032247, 324609913, 361705873, 417684523, 449364197, 451995587, 454052213, 466037563, 536504713, 574746467, 596095613

Offset: 1

Giovanni Resta, Feb 27 2006

nonn

			p = 306167, f(p) = 93738231893, f(f(p)) = 8786856118425842363453, f(f(f(p))) = 77208840445917661077402487029419236950083213 and the 88-digit number f(f(f(f(p)))) are all prime numbers.

Cf. A062324, A116886, A116887, A116889.

Select[Prime[Range[9! ]],PrimeQ[ #^2+4]&&PrimeQ[(#^2+4)^2+4]&&PrimeQ[((#^2+4)^2+4)^2+4]&&PrimeQ[(((#^2+4)^2+4)^2+4)^2+4]&] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2010 *)
p4Q[p_]:=AllTrue[NestList[#^2+4&,p,4],PrimeQ]; Select[Prime[Range[312*10^5]],p4Q] (* Harvey P. Dale, Nov 20 2023 *)

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]&]

A132260 Array T(k,n) = n-th prime p such that 2^2^k + p^2^k is prime, k>2, read by antidiagonals.

Original entry on oeis.org

13, 89, 137, 29, 107, 223, 37, 59, 127, 331, 113, 53, 101, 139, 389, 113, 223, 181, 103, 173, 491, 13, 1223, 5279, 491, 109, 179, 563, 1151, 181, 1277, 7517, 547, 181, 229, 647, 43, 2153, 761, 1993, 8039, 619, 199, 233, 701, 53, 271, 3559, 4133, 2399, 9833, 661, 379, 349, 773

Offset: 3

Jonathan Vos Post, Aug 15 2007

These were computed by Ignacio Larrosa Cañestro, who cautions that some are only probable primes. The k=3 row is A157950. The main diagonal is A132261.

			The array begins:
   n  |   1    2    3    4    5    6     7     8     9    10
  ----+--------------------------------------------------------
  k=3 |  13  137  223  331  389  491   563   647   701   773
  k=4 |  89  107  127  139  173  179   229   233   349   421
  k=5 |  29   59  101  103  109  181   199   379   769   881
  k=6 |  37   53  181  491  547  619   661   677   911   941
  k=7 | 113  223 5279 7517 8039 9833 12197 13757 21467 23447
  k=8 | 113 1223 1277 1993 2399 9349  9739 10211 10973 11059

Cf. A000040, A062324, A132261, A157764, A157950, A158361, A158477.

More terms from Jinyuan Wang, Feb 01 2022

A157950 Primes p such that p^8 + 2^8 is prime.

Original entry on oeis.org

13, 137, 223, 331, 389, 491, 563, 647, 701, 773, 797, 1063, 1181, 1531, 1579, 1811, 2027, 2087, 2269, 2333, 2393, 2617, 2687, 2699, 2857, 3313, 3467, 3623, 3637, 3691, 3739, 3761, 3863, 3877, 4133, 4201, 4283, 4297, 4877, 5023, 5839, 5897, 6043, 6053

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 10 2009

nonn

17 divides p^8 + 2^8 if k is odd and p = 17k +- 6, 17k +- 10, 17k +- 12, 17k +- 14, so only 8 integers p in each interval of length 34 need to be tested for the primality of p and of p^8 + 2^8: those of the forms p = 17k +- 2 (which yield terms 223, 389, 491, 563, 797, 1579, 3313, 3623, 3691, ...), p = 17k +- 4 (which yield terms 13, 701, 2027, 2087, 2333, 2393, 2699, ...), p = 17k +-8 (which yield terms 331, 773, 1063, 1181, 1811, 2269, ...), and p = 17k +-16 (which yield terms 137, 647, 1531, 2617, 2687, 2857, 3467, 3637, ...).

It is conjectured that this sequence is infinite.

			n=11: 11^8 + 2^8 = 214359137 = 17 * 241 * 52321, not prime, so 11 is not a term;
n=13: 13^8 + 2^8 = 815730977 is prime, so 13 is a term.

Leonard E. Dickson, History of the Theory of Numbers.
Richard Guy, Unsolved Problems in Number Theory.

Cf. A062324, A157764.

a := proc (n) if isprime(ithprime(n)^8+256) = true then ithprime(n) else end if end proc: seq(a(n), n = 1 .. 900); # Emeric Deutsch, Mar 14 2009

Definition corrected by Emeric Deutsch, Mar 14 2009
Extended by Emeric Deutsch, Mar 14 2009
Edited by Jon E. Schoenfield, Jan 29 2019

A263721 The prime p in the Fouvry-Iwaniec prime k^2 + p^2 (A185086), or the larger of k and p if both are prime.

Original entry on oeis.org

2, 3, 5, 5, 7, 5, 3, 5, 3, 7, 11, 7, 11, 13, 7, 2, 13, 13, 5, 17, 13, 11, 5, 17, 7, 17, 19, 3, 17, 7, 19, 5, 11, 19, 13, 23, 5, 17, 19, 13, 23, 5, 2, 19, 17, 11, 5, 23, 29, 29, 23, 19, 29, 13, 31, 31, 23, 11, 3, 5, 31, 13, 2, 29, 5, 13, 31, 2, 11, 5, 31, 37, 23, 37, 3, 7, 23, 3, 13, 31, 19, 37, 41, 11

Offset: 1

Jonathan Sondow and Robert G. Wilson v, Oct 24 2015

nonn

The sequence is well-defined by the uniqueness part of Fermat's two-squares theorem.

The sequence is infinite, since Fouvry and Iwaniec proved that A185086 is infinite.

			A185086(2) = 13 = 2^2 + 3^2 and A185086(6) = 61 = 5^2 + 6^2, so a(2) = 3 and a(6) = 5.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Étienne Fouvry and Henryk Iwaniec, Gaussian primes, Acta Arithmetica 79:3 (1997), pp. 249-287.

Cf. A002313, A002144, A028916, A045637, A062324, A185086, A240130, A262340, A263722.

p = 2; lst = {}; While[p < 100, k = 1; While[k < 101, If[PrimeQ[k^2 + p^2], AppendTo[lst, {k^2 + p^2, If[PrimeQ@ k, Max[k, p], p]}]]; k++]; p = NextPrime@ p]; Transpose[Union@ lst][[2]]

do(lim)=my(v=List(),p2,t); forprime(p=2, sqrtint(lim\=1), p2=p^2; for(k=1, sqrtint(lim-p2), if(isprime(t=p2+k^2), listput(v, [t, if(isprime(k),max(k,p),p)])))); v=vecsort(Set(v),1); apply(u->u[2], v) \\ Charles R Greathouse IV, Aug 21 2017

a(n)^2 = A185086(n) - k^2 for some integer k > 0.

A158477 Primes p with property that Q(p) = p^32+2^32 is prime.

Original entry on oeis.org

29, 59, 101, 103, 109, 181, 199, 379, 769, 881, 919, 977, 1097, 1213, 1303, 1583, 2099, 2113, 2441, 2521, 2617, 2777, 3067, 3739, 4133, 4289, 4519, 4931, 5039, 5113, 5227, 5417, 5743, 5783, 6143, 6373, 6691, 8053, 8209, 8287, 8513, 9109, 9203, 9689, 9787, 9923, 9941

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 20 2009

nonn

1) Q=(p^16)^2+(2^16)^2 only for Q=4k+1 because of Fermat/Euler/Lagrange theorem concerning prime as sum of two squares.
2) It is conjectured that sequence a(n) is infinite.
3) Note the twin prime: a(3)=101, a(4)=103.
The next set of twins is a(101)=30557, a(102)=30559. - Robert Israel, Apr 05 2016

			p=3: 3^32+2^32=1853024483819137 = 1153 x 1607133116929 no prime;
also for following primes p=5, 7, 11, 13, 17, 19, 23: Q(p) no prime;
p=29: 29^32+2^32=62623297589448778360828428329074752313100292737 is prime => a(1)=29.

Richard E. Crandall, Carl Pomerance, Prime Numbers: A Computational Perspective, Springer 2001.
Leonard E. Dickson, History of the Theory of Numbers, Dover Pub. Inc., 2005.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A062324, A157764, A157950, A158361.

select(t -> isprime(t) and isprime(t^32 + 2^32), [seq(i,i=3..10000,2)]); # Robert Israel, Apr 05 2016

With[{c=2^32},Select[Prime[Range[1300]],PrimeQ[#^32+c]&]] (* Harvey P. Dale, May 04 2018 *)

isA158477(n) = isprime(n) && isprime(n^32+4294967296) \\ Michael B. Porter, Dec 17 2009

lista(nn) = forprime(p=3, nn, if(ispseudoprime(p^32+2^32), print1(p, ", "))); \\ Altug Alkan, Apr 05 2016

n^32+2^32 and n to be prime.

A357426 Primes p such that p^2+4 is a prime times 5^k for some k >= 1.

Original entry on oeis.org

11, 19, 31, 41, 61, 71, 79, 89, 109, 131, 139, 149, 151, 181, 191, 239, 241, 251, 379, 389, 409, 421, 461, 499, 509, 541, 599, 631, 659, 661, 709, 719, 769, 811, 919, 1009, 1019, 1021, 1031, 1109, 1129, 1151, 1201, 1231, 1291, 1361, 1399, 1409, 1451, 1489, 1549, 1601, 1621, 1721, 1789, 1871, 1889, 1931, 2011, 2039, 2069, 2131, 2179, 2221, 2251, 2309, 2341, 2351

Offset: 1

J. M. Bergot and Robert Israel, Sep 27 2022

nonn

All terms == 1 or 9 (mod 10).

			a(4) = 41 is a term because 41 is prime and 41^2+4 = 1685 = 337 * 5^1 where 337 is prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Disjoint from A062324.

filter:= proc(p) local v;
  if not isprime(p) then return false fi;
  v:= p^2+4;
  isprime(v/5^padic:-ordp(v,5))
end proc:
filter(11):= true:
select(filter, [seq(seq(10*i+j, j= [1,9]),i=1..1000)]);

q[p_] := (e = IntegerExponent[m = p^2 + 4, 5]) > 0 && (m==5^e || PrimeQ[m/5^e]); Select[Prime[Range[350]], q] (* Amiram Eldar, Sep 28 2022 *)

A153645 Primes p such that p^2 + 4 and p^2 + 4p + 2 are also prime.

Original entry on oeis.org

3, 5, 7, 13, 17, 47, 67, 73, 137, 167, 277, 307, 313, 487, 503, 593, 607, 613, 787, 823, 1117, 1123, 1237, 1523, 1543, 1637, 1987, 2777, 2887, 3037, 3163, 3433, 3457, 3463, 3797, 3853, 4093, 4283, 4583, 5113, 5297, 5323, 5683, 5953, 6047, 6577, 6803, 6823

Offset: 1

Vincenzo Librandi, Dec 30 2008

Subsequence of A062324.

			For prime p = 3, p^2+4 = 13 and p^2+4p+2 = 23 are prime; for p = 67, p^2+4 = 4493 and p^2+4p+2 = 4759 are prime.

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

Cf. A062324 (p and p^2+4 are both prime).

[ p: p in PrimesUpTo(7000) | IsPrime(p^2+4) and IsPrime(p^2+4*p+2) ];

a := proc (n) if isprime(n) = true and isprime(n^2+4) = true and isprime(n^2+4*n+2) = true then n else end if end proc: seq(a(n), n = 1 .. 7000); # Emeric Deutsch, Jan 02 2009

Select[Prime[Range[10000]],PrimeQ[#^2+4]&&PrimeQ[#^2 +4#+2]&] (* Vincenzo Librandi, Jul 27 2012 *)

Edited, corrected (three terms deleted) and extended beyond a(10) by Klaus Brockhaus, Jan 02 2009

Corrected and extended by Emeric Deutsch, Jan 02 2009

cp's OEIS Frontend

A165218 Primes q of the form q=p^2+4 (p=prime) such that r=q^2+4 is also prime.

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A116888 Primes p that remain prime through 4 iterations of the function f(p) = p^2 + 4.

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

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Programs

Mathematica

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

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Programs

Mathematica

A132260 Array T(k,n) = n-th prime p such that 2^2^k + p^2^k is prime, k>2, read by antidiagonals.

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A157950 Primes p such that p^8 + 2^8 is prime.

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A263721 The prime p in the Fouvry-Iwaniec prime k^2 + p^2 (A185086), or the larger of k and p if both are prime.

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A158477 Primes p with property that Q(p) = p^32+2^32 is prime.

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