A002384 -id:A002384 - cp's OEIS frontend

A075716 1+n+n^s is a prime, s=15.

1, 2, 30, 32, 35, 54, 62, 77, 101, 120, 138, 161, 171, 186, 210, 234, 269, 285, 311, 341, 362, 368, 374, 467, 476, 486, 531, 567, 578, 720, 737, 740, 780, 806, 824, 932, 990, 1035, 1037, 1041, 1049, 1067, 1089, 1136, 1137, 1146, 1167, 1202, 1251, 1269

Offset: 1

Zak Seidov, Oct 03 2002

For s = 5,8,11,14,17,20,..., n_s=1+n+n^s is always composite for any n>1. Also at n=1, n_s=3 is a prime for any s. So it is interesting to consider only the cases of s =/= 5,8,11,14,17,20,... and n>1. Here i consider the case s=15 and find several first n's making n_s a prime (or a probable prime).

			2 is OK because at s=15, n=2, n_s=1+n+n^s=32771 is a prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Cf. A002384, A075715, A075717.

[n: n in [0..2000] | IsPrime(s) where s is 1+n+n^15]; // Vincenzo Librandi, Jul 28 2014

Select[Range[1500], PrimeQ[1 + # + #^15] &] (* Harvey P. Dale, Dec 13 2010 *)
Select[Range[2000], PrimeQ[Total[#^Range[1, 15, 14]] + 1] &] (* Vincenzo Librandi, Jul 28 2014 *)

for(n=1,1000,if(isprime(1+n+n^15),print1(n",")))

More terms from Ralf Stephan, Apr 05 2003

A075717 1+n+n^13 is a prime.

Original entry on oeis.org

1, 5, 15, 39, 50, 56, 105, 116, 128, 153, 168, 170, 243, 245, 264, 308, 314, 369, 401, 429, 480, 489, 531, 551, 599, 608, 680, 690, 699, 701, 785, 804, 939, 978, 1050, 1056, 1065, 1073, 1110, 1169, 1224, 1226, 1271, 1283, 1308, 1310, 1391, 1401

Offset: 1

Zak Seidov, Oct 03 2002

For s = 5,8,11,14,17,20,..., n_s=1+n+n^s is always composite for any n>1. Also at n=1, n_s=3 is a prime for any s. So it is interesting to consider only the cases of s =/= 5,8,11,14,17,20,... and n>1. Here I consider the case s=13 and find several first n's making n_s a prime (or a probable prime).

			5 is OK because at s=13, n=2, n_s=1+n+n^s=1220703131 is a prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1900

Cf. A002384, A075716, A075718.

[n: n in [0..2000] | IsPrime(s) where s is 1+n+n^13]; // Vincenzo Librandi, Jul 28 2014

Select[Range[1500], PrimeQ[1 + # + #^13] &] (* Harvey P. Dale, Apr 20 2013 *)

n=0; for(k=1, 60, n=n+1; while(!isprime(1+n+n^13), n=n+1); print1(n","))

More terms from Ralf Stephan, Mar 20 2003

A075718 1+n+n^s is a prime, s=12.

Original entry on oeis.org

1, 2, 14, 30, 32, 44, 65, 87, 90, 122, 134, 149, 162, 186, 189, 227, 237, 249, 255, 266, 311, 354, 366, 456, 476, 485, 561, 567, 584, 597, 605, 650, 665, 672, 689, 720, 771, 819, 884, 899, 975, 990, 1059, 1082, 1092, 1191, 1200, 1241, 1257, 1295, 1347, 1367

Offset: 1

Zak Seidov, Oct 03 2002

For s = 5,8,11,14,17,20,..., n_s=1+n+n^s is always composite for any n>1. Also at n=1, n_s=3 is a prime for any s. So it is interesting to consider only the cases of s =/= 5,8,11,14,17,20,... and n>1. Here i consider the case s=12 and find several first n's making n_s a prime (or a probable prime).

			2 is OK because at s=12, n=2, n_s=1+n+n^s=4099 is a prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..2200

Cf. A002384, A075717, A075719.

[n: n in [0..2000] | IsPrime(s) where s is 1+n+n^12]; // Vincenzo Librandi, Jul 28 2014

Select[Range[2000], PrimeQ[1 + # + #^12] &] (* Vincenzo Librandi, Jul 28 2014 *)

for(n=1,1000,if(isprime(1+n+n^12),print1(n",")))

More terms from Ralf Stephan, Apr 05 2003

A075719 1+n+n^s is a prime, s=10.

Original entry on oeis.org

1, 3, 8, 21, 23, 26, 33, 36, 38, 45, 51, 57, 69, 71, 78, 92, 107, 117, 149, 156, 170, 176, 179, 195, 209, 216, 219, 224, 261, 293, 321, 341, 359, 374, 378, 386, 390, 404, 410, 413, 420, 474, 492, 507, 516, 546, 569, 572, 582, 621, 632, 683, 767, 783, 789, 809

Offset: 1

Zak Seidov, Oct 03 2002

For s = 5,8,11,14,17,20,..., n_s=1+n+n^s is always composite for any n>1. Also at n=1, n_s=3 is a prime for any s. So it is interesting to consider only the cases of s =/= 5,8,11,14,17,20,... and n>1. Here i consider the case s=10 and find several first n's making n_s a prime (or a probable prime).

			3 is OK because at s=10, n=3, n_s=1+n+n^s=59053 is a prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..3300

Cf. A002384, A075718, A075720.

[n: n in [0..1000] | IsPrime(s) where s is 1+n+n^10]; // Vincenzo Librandi, Jul 28 2014

Select[Range[1000], PrimeQ[1 + # + #^10] &] (* Vincenzo Librandi, Jul 28 2014 *)

for(n=1,1000,if(isprime(1+n+n^10),print1(n",")))

More terms from Ralf Stephan, Apr 05 2003

A075720 Numbers n such that n^9 + n + 1 is a prime.

Original entry on oeis.org

1, 3, 9, 11, 14, 15, 18, 23, 38, 51, 66, 89, 95, 140, 170, 185, 186, 194, 239, 258, 294, 315, 345, 366, 384, 386, 393, 401, 404, 408, 429, 459, 485, 495, 506, 531, 573, 611, 614, 665, 675, 678, 680, 683, 695, 750, 771, 791, 849, 870, 879, 941, 954, 1016, 1086

Offset: 1

Zak Seidov, Oct 03 2002

For s = 5,8,11,14,17,20,..., n_s=1+n+n^s is always composite for any n>1. Also at n=1, n_s=3 is a prime for any s. So it is interesting to consider only the cases of s != 5,8,11,14,17,20,... and n>1. Here I consider the case s=9 and find several first n's making n_s a prime (or a probable prime).

			9 is a member because at n=9, 1 + n + n^s = 387420499 is a prime.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A002384, A075719, A075722.

[n: n in [0..1500] | IsPrime(s) where s is 1+n+n^9]; // Vincenzo Librandi, Jul 28 2014

Select[Range[1500], PrimeQ[1 + # + #^9] &] (* Vincenzo Librandi, Jul 28 2014 *)

for(n=1,10^3,if(isprime(n^9+n+1),print1(n,", "))) \\ Derek Orr, Feb 07 2015

A088503 Numbers n such that (n^2 + 3)/4 is prime.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 25, 29, 31, 35, 41, 43, 49, 55, 67, 77, 83, 101, 109, 115, 119, 125, 133, 139, 143, 151, 155, 157, 161, 179, 181, 199, 203, 211, 221, 223, 235, 239, 263, 277, 283, 287, 295, 301, 307, 311, 323, 325, 329, 335, 337, 347, 353, 377, 379, 385

Offset: 1

Pierre CAMI, Nov 13 2003

Under Bunyakovsky's conjecture this sequence is infinite. - Charles R Greathouse IV, Dec 28 2011

			(25*25 + 3)/4 = 157, 157 is prime, 25 is the 7th term of the sequence.

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

Cf. A002383, A002384, A110284.

[n: n in [1..400 by 2] | IsPrime((n^2 + 3) div 4)]; // Vincenzo Librandi, Oct 06 2012

Select[Range[500], PrimeQ[(#^2 + 3)/4] &] (* Vincenzo Librandi, Oct 06 2012 *)

for(k=2,1e3,if(isprime(k^2+k+1),print1(2*k+1", "))) \\ Charles R Greathouse IV, Dec 28 2011

a(n) = 2*A002384(n) + 1 = sqrt(A110284(n)). - Ray Chandler, Sep 07 2005

Corrected and extended by Ray Chandler, Nov 16 2003

A075713 1+n+n^19 is a prime.

Original entry on oeis.org

1, 8, 15, 20, 29, 33, 48, 98, 105, 114, 177, 231, 260, 302, 320, 338, 387, 393, 432, 456, 473, 488, 489, 558, 564, 632, 677, 680, 726, 770, 795, 828, 855, 869, 1019, 1026, 1050, 1056, 1079, 1119, 1124, 1217, 1266, 1302, 1373, 1454, 1467, 1547

Offset: 1

Zak Seidov, Oct 03 2002

For s = 5,8,11,14,17,20,..., n_s=1+n+n^s is always composite for any n>1. Also at n=1, n_s=3 is a prime for any s. So it is interesting to consider only the cases of s =/= 5,8,11,14,17,20,... and n>1. Here i consider the case s=19 and find several first n's making n_s a prime (or a probable prime).

			8 is OK because at s=19, n=8, n_s=1+n+n^s=144115188075855881 is a prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1700

Cf. A002384, A075714.

[n: n in [0..1600] | IsPrime(s) where s is 1+n+n^19]; // Vincenzo Librandi, Jul 28 2014

Select[Range[2000], PrimeQ[1 + # + #^19] &] (* Vincenzo Librandi, Jul 28 2014 *)

for(n=1, 2000, if(isprime(1+n+n^19), print1(n",")))

More terms from Ralf Stephan, Mar 31 2003

A075722 Numbers n such that 1 + n + n^s is a prime, s = 7.

Original entry on oeis.org

1, 2, 15, 21, 29, 39, 42, 53, 57, 77, 81, 92, 117, 123, 131, 147, 149, 153, 167, 168, 200, 204, 207, 233, 249, 251, 252, 275, 278, 314, 317, 326, 357, 372, 378, 380, 410, 422, 434, 438, 440, 462, 467, 468, 498, 516, 546, 585, 587, 596, 608, 615, 621, 636

Offset: 1

Zak Seidov, Oct 03 2002

For s = 5,8,11,14,17,20,..., n_s=1+n+n^s is always composite for any n>1. Also at n=1, n_s=3 is a prime for any s. So it is interesting to consider only the cases of s != 5,8,11,14,17,20,... and n>1. Here I consider the case s=7 and find several first n's making n_s a prime (or a probable prime).

			15 is OK because at s=7, n=15, n_s = 1 + n + n^s = 170859391 is a prime.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A002384, A075720, A075723.

[n: n in [0..1000] | IsPrime(s) where s is 1+n+n^7]; // Vincenzo Librandi, Jul 28 2014

Select[Range[1000], PrimeQ[1 + # + #^7] &] (* Vincenzo Librandi, Jul 28 2014 *)

for(n=1,10^3,if(isprime(n^7+n+1),print1(n,", "))) \\ Derek Orr, Feb 07 2015

More terms from Ralf Stephan, Apr 05 2003

A096176 Numbers k such that (k^3-1)/(k-1) is prime.

Original entry on oeis.org

2, 3, 5, 6, 8, 12, 14, 15, 17, 20, 21, 24, 27, 33, 38, 41, 50, 54, 57, 59, 62, 66, 69, 71, 75, 77, 78, 80, 89, 90, 99, 101, 105, 110, 111, 117, 119, 131, 138, 141, 143, 147, 150, 153, 155, 161, 162, 164, 167, 168, 173, 176, 188, 189, 192, 194, 203, 206, 209, 215, 218

Offset: 1

Hugo Pfoertner, Jun 22 2004

nonn

Numbers k > 1 such that k^2 + k + 1 is a prime. - Vincenzo Librandi, Nov 16 2010

Therefore essentially the same as A002384. - Georg Fischer, Oct 06 2018

			a(5) = 8 because (8^3-1)/(8-1) = 511/7 = 73 is prime.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A096174 (n^3+1)/(n+1) is prime, A081257 largest prime factor of n^3-1, A096175 n^3-1 is an odd semiprime.
Cf. A028491, A004061. - Daniel McCandless (dkmccandless(AT)gmail.com), Aug 31 2009
Cf. A002384.

[n: n in [2..220] |IsPrime((n^3-1) div (n -1))]; // Vincenzo Librandi, Oct 07 2018

Select[Range[2,550],PrimeQ[(#^3-1)/(#-1)]&] (* Harvey P. Dale, Sep 10 2009 *)

is(n)=isprime(n^2+n+1) \\ Charles R Greathouse IV, Jun 05 2017

3 and 5 added by Daniel McCandless (dkmccandless(AT)gmail.com), Aug 31 2009

Corrected terms, including many previously omitted terms, from Harvey P. Dale, Sep 10 2009

A247244 Smallest prime p such that (n^p + (n+1)^p)/(2n+1) is prime, or -1 if no such p exists.

Original entry on oeis.org

3, 3, 3, 5, 3, 3, 7, 3, 7, 53, 47, 3, 7, 3, 3, 41, 3, 5, 11, 3, 3, 11, 11, 3, 5, 103, 3, 37, 17, 7, 13, 37, 3, 269, 17, 5, 17, 3, 5, 139, 3, 11, 78697, 5, 17, 3671, 13, 491, 5, 3, 31, 43, 7, 3, 7, 2633, 3, 7, 3, 5, 349, 3, 41, 31, 5, 3, 7, 127, 3, 19, 3, 11, 19, 101, 3, 5, 3, 3

Offset: 1

Eric Chen, Nov 28 2014

nonn

All terms are odd primes.
a(79) > 10000, if it exists.
a(80)..a(93) = {3, 7, 13, 7, 19, 31, 13, 163, 797, 3, 3, 11, 13, 5}, a(95)..a(112) = {5, 2657, 19, 787, 3, 17, 3, 7, 11, 1009, 3, 61, 53, 2371, 5, 3, 3, 11}, a(114)..a(126) = {103, 461, 7, 3, 13, 3, 7, 5, 31, 41, 23, 41, 587}, a(128)..a(132) = {7, 13, 37, 3, 23}, a(n) is currently unknown for n = {79, 94, 113, 127, 133, ...} (see the status file under Links).

			a(10) = 53 because (10^p + 11^p)/21 is composite for all p < 53 and prime for p = 53.

Aurelien Gibier, Table of n, a(n) for n = 1..78 (first 42 terms from Robert G. Wilson v)
Robert G. Wilson v, Table of n, a(n) for n = 1..1000 with status of unknown terms [updated/edited by Jon E. Schoenfield]

Cf. A058013, A125713, A000978, A057469, A128066, A128335, A128336, A187805, A181141, A187819, A217095, A185239, A213216, A225097, A224984, A221637, A227170, A228573, A227171, A225818, A227172, A227173, A227174.

lmt = 4200; f[n_] := Block[{p = 2}, While[p < lmt && !PrimeQ[((n + 1)^p + n^p)/(2n + 1)], p = NextPrime@ p]; If[p > lmt, 0, p]]; Do[Print[{n, f[n] // Timing}], {n, 1000}] (* Robert G. Wilson v, Dec 01 2014 *)

a(n)=forprime(p=3, , if(ispseudoprime((n^p+(n+1)^p)/(2*n+1)), return(p)))

a(n) = 3 if and only if n^2 + n + 1 is a prime (A002384).

a(43) from Aurelien Gibier, Nov 27 2023

cp's OEIS Frontend

A075716 1+n+n^s is a prime, s=15.

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A075717 1+n+n^13 is a prime.

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A075718 1+n+n^s is a prime, s=12.

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A075719 1+n+n^s is a prime, s=10.

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A075720 Numbers n such that n^9 + n + 1 is a prime.

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A088503 Numbers n such that (n^2 + 3)/4 is prime.

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A075713 1+n+n^19 is a prime.