A007591 -id:A007591 - cp's OEIS frontend

A122062 Numbers k such that k^2 + 16 is prime.

1, 5, 9, 11, 15, 21, 25, 29, 31, 41, 49, 51, 55, 65, 75, 79, 81, 89, 91, 95, 99, 109, 115, 119, 121, 125, 129, 151, 165, 179, 191, 211, 219, 221, 229, 231, 245, 249, 265, 275, 281, 289, 291, 295, 299, 301, 311, 315, 335, 351, 355, 361, 365, 369, 381, 389, 391

Offset: 1

Parthasarathy Nambi, Sep 14 2006

nonn

			If k=99 then k^2 + 16 = 9817 (prime).

Daniel Starodubtsev, Table of n, a(n) for n = 1..5000

Cf. A005574, A067201, A049422, A007591, A078402, A114269, A114271, A114272, A114273, A114274, A114275, A113536, A121250, A121982.

[n: n in [0..6000] | IsPrime(n^2 + 16)]; // Vincenzo Librandi, Nov 13 2010

Select[Range[1,501,2],PrimeQ[16+#^2]&] (* Harvey P. Dale, Dec 04 2018 *)

is(n)=isprime(n^2+16) \\ Charles R Greathouse IV, Jun 06 2017

A246562 Primes p such that 4+p, 4+p^2, 4+p^3, 4+p^5, and 4+p^7 are all prime.

Original entry on oeis.org

7, 469363, 2552713, 3378103, 6595597, 6629683, 39837517, 46024063, 46167307, 97371007, 97629403, 105528217, 136983307, 169483033, 202953613, 213792193, 216520987, 216738043, 221705647, 304033927, 317502193, 359133553

Offset: 1

Zak Seidov, Aug 29 2014

nonn

All terms are == {3, 7} mod 10. Subsequence of A246519.

Zak Seidov, Table of n, a(n) for n = 1..60

Cf. A007591, A073573, A125260, A172367, A246519.

Select[Prime[Range[193*10^5]],AllTrue[#^{1,2,3,5,7}+4,PrimeQ]&] (* Harvey P. Dale, Sep 07 2024 *)

forprime(p=1,10^9,if(ispseudoprime(4+p) && ispseudoprime(4+p^2) && ispseudoprime(4+p^3) && ispseudoprime(4+p^5) && ispseudoprime(4+p^7), print1(p,", "))) \\ Derek Orr, Aug 30 2014

A129412 Numbers k such that mean of 7 consecutive squares starting with k^2 is prime.

Original entry on oeis.org

0, 2, 4, 10, 12, 14, 24, 30, 32, 34, 42, 44, 54, 62, 64, 70, 82, 84, 92, 94, 100, 112, 114, 122, 132, 134, 144, 152, 160, 164, 174, 180, 190, 200, 204, 212, 214, 230, 232, 240, 242, 250, 252, 262, 264, 272, 274, 284, 290, 300, 304, 310, 314, 344, 354, 370, 372

Offset: 1

Zak Seidov, Apr 14 2007

Sum of 7 consecutive squares starting with k^2 is equal to 7*(13 + 6*k + k^2) and mean is (13 + 6*k + k^2) = (k+3)^2+4. Hence a(n) = A007591(n+1)-3.

			(0^2+...+6^2)/7=13 prime, (2^2+...+8^2)/7=29 prime, (4^2+...+10^2)/7=53 prime.

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

Cf. A005473, A056899, A067201, A007591, A129389, A098062.

Select[Range[0,400],PrimeQ[Mean[Range[#,#+6]^2]]&]  (* Harvey P. Dale, Apr 23 2011 *)

is(n)=isprime(n^2+6*n+13) \\ Charles R Greathouse IV, Jun 13 2017

A264790 Numbers k such that k^2 + 17 is prime.

Original entry on oeis.org

0, 6, 24, 60, 66, 78, 90, 108, 144, 162, 174, 186, 234, 252, 294, 300, 318, 330, 336, 342, 372, 396, 420, 438, 456, 462, 468, 498, 528, 594, 636, 648, 654, 672, 720, 750, 798, 804, 834, 858, 888, 924, 930, 966, 984, 990, 1014, 1026, 1032, 1086, 1158, 1194, 1200

Offset: 1

Ilya Gutkovskiy, Nov 25 2015

nonn

Primes of the form k^2 + 17 have a representation as a sum of 2 squares because they belong to A002144.

All terms are multiple of 6.

			a(3) = 24 because 24^2 + 17 = 593, which is prime.

Daniel Starodubtsev, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Near-Square Prime

Cf. A228244 (associated primes).

Other sequences of the type "Numbers n such that n^2 + k is prime": A005574 (k=1), A067201 (k=2), A049422 (k=3), A007591 (k=4), A078402 (k=5), A114269 (k=6), A114270 (k=7), A114271 (k=8), A114272 (k=9), A114273 (k=10), A114274 (k=11), A114275 (k=12), A113536 (k=13), A121250 (k=14), A121982 (k=15), A122062 (k=16).

[n: n in [0..1200 ] | IsPrime(n^2+17)]; // Vincenzo Librandi, Nov 25 2015

Select[Range[0, 1200], PrimeQ[#^2 + 17] &] (* Michael De Vlieger, Nov 25 2015 *)

for(n=0, 1e3, if(isprime(n^2+17), print1(n, ", "))) \\ Altug Alkan, Nov 25 2015

A000005(A241847(a(n))) = 2.

A241847(a(n)) = A228244(n).

Edited by Bruno Berselli, Nov 26 2015

A243095 Least integer m > 1 such that 4 + m^n is prime or 1 if only 4 + 1^n is prime.

Original entry on oeis.org

3, 3, 3, 1, 7, 3, 7, 1, 3, 3, 9, 1, 33, 7, 9, 1, 43, 17, 27, 1, 9, 3, 7, 1, 55, 47, 285, 1, 27, 3, 39, 1, 43, 117, 163, 1, 63, 255, 15, 1, 87, 3, 43, 1, 187, 77, 37, 1, 105, 45, 25, 1, 99, 305, 79, 1, 3, 27, 903, 1, 127, 293, 255, 1, 27, 27, 435, 1, 207, 143, 127, 1, 117, 295, 1159, 1, 477

Offset: 1

Zak Seidov, Aug 29 2014

nonn

If n is a multiple of 4 then 4 + m^n is prime iff m = 1.

4 + m^(4*x) = (m^(2*x)-2*m^x+2) * (m^(2*x)+2*m^x+2). - Jens Kruse Andersen, Sep 02 2014

Zak Seidov and Jens Kruse Andersen, Table of n, a(n) for n = 1..1000 (first 200 terms from Zak Seidov)

Cf. A007591, A073573, A125260, A172367, A246519.

a(n)=if(n%4==0,return(1));m=2;while(!ispseudoprime(4+m^n),m++);return(m)
vector(100,n,a(n)) \\ Derek Orr, Aug 29 2014

A284014 Numbers k such that {k + 2, k + 4} and {k^2 + 2, k^2 + 4} are both twin prime pairs.

Original entry on oeis.org

1, 3, 15, 57, 147, 2085, 6687, 6957, 11055, 15267, 17385, 17577, 20505, 20637, 23667, 26247, 31077, 31317, 32115, 32967, 34497, 39225, 47775, 52065, 53715, 55335, 56205, 58365, 62187, 63585, 66567, 67215, 70875, 77235, 77475, 82005, 85827, 89595, 89817, 107505

Offset: 1

K. D. Bajpai, Mar 18 2017

nonn

After a(1), all the terms are multiples of 3.

After a(2), all the terms are congruent to 5 or 7 (mod 10).

			a(2) = 3, {3 + 2 = 5, 3 + 4 = 7} and {3^2 + 2 = 11, 3^2 + 4 = 13} are twin prime pairs.
a(3) = 15, {15 + 2 = 17, 15 + 4 = 19} and {15^2 + 2 = 227, 15^2 + 4 = 229} are twin prime pairs.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Appears to be the intersection of A086381 and A256388, but that may be unproven.

Cf. A000040, A001359, A007591, A010051, A067201, A178336, A178337.

[n: n in [0..100000] | IsPrime(n+2) and IsPrime(n+4) and IsPrime(n^2+2) and IsPrime(n^2+4)];

Select[Range[1000000], PrimeQ[# + 2] && PrimeQ[# + 4] && PrimeQ[#^2 + 2] && PrimeQ[#^2 + 4] &]

for(n=1, 100000,2; if(isprime(n+2) && isprime(n+4) && isprime(n^2+2) &&isprime(n^2+4), print1(n, ", ")))

;; With Antti Karttunen's IntSeq-library.
(define A284014 (MATCHING-POS 1 1 (lambda (n) (and (= 1 (A010051 (+ n 2))) (= 1 (A010051 (+ n 4))) (= 1 (A010051 (+ (* n n) 2))) (= 1 (A010051 (+ (* n n) 4)))))))
;; Antti Karttunen, Apr 15 2017

A089747 Numbers n such that n^2 - 2n + 5 is prime.

Original entry on oeis.org

0, 2, 4, 6, 8, 14, 16, 18, 28, 34, 36, 38, 46, 48, 58, 66, 68, 74, 86, 88, 96, 98, 104, 116, 118, 126, 136, 138, 148, 156, 164, 168, 178, 184, 194, 204, 208, 216, 218, 234, 236, 244, 246, 254, 256, 266, 268, 276, 278, 288, 294, 304, 308, 314, 318, 348, 358, 374

Offset: 1

Giovanni Teofilatto, Jan 08 2004

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna, 1988.
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta, UTET, CittaStudiEdizioni, Milano, 1997.

Cf. A005473 gives primes, A007591.

[n: n in [0..100]|IsPrime(n^2-2*n+5)] // Vincenzo Librandi, Nov 16 2010

Select[Range[0,400,2],PrimeQ[#^2-2#+5]&] (* Harvey P. Dale, Jan 04 2015 *)

is(n)=isprime(n^2-2*n+5) \\ Charles R Greathouse IV, Jun 13 2017

a(n) = A007591(n-1) + 1 for n > 1. [corrected by Georg Fischer, Jun 20 2020]

cp's OEIS Frontend

A122062 Numbers k such that k^2 + 16 is prime.

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A246562 Primes p such that 4+p, 4+p^2, 4+p^3, 4+p^5, and 4+p^7 are all prime.

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A129412 Numbers k such that mean of 7 consecutive squares starting with k^2 is prime.

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A264790 Numbers k such that k^2 + 17 is prime.

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A243095 Least integer m > 1 such that 4 + m^n is prime or 1 if only 4 + 1^n is prime.

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A284014 Numbers k such that {k + 2, k + 4} and {k^2 + 2, k^2 + 4} are both twin prime pairs.

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A089747 Numbers n such that n^2 - 2n + 5 is prime.

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