A153504 -id:A153504 - cp's OEIS frontend

A155211 Numbers n such that n^4+(n+1)^4 is a prime.

1, 2, 3, 4, 6, 8, 9, 12, 13, 14, 16, 25, 26, 27, 31, 33, 34, 36, 37, 38, 40, 43, 48, 54, 63, 67, 68, 72, 74, 78, 82, 87, 88, 89, 97, 98, 104, 105, 109, 110, 111, 119, 121, 122, 123, 129, 145, 156, 157, 162, 163, 166, 167, 172, 173, 179, 180, 182, 184, 186, 187, 189, 195

Offset: 1

Vincenzo Librandi, Jan 22 2009

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

Cf. A153504, A154535.

[n: n in [1..200] |  IsPrime(n^4+(n+1)^4)]; // Vincenzo Librandi, Aug 31 2012

f[n_]:=n^4+(n+1)^4;lst={};Do[a=f[n];If[PrimeQ[a],AppendTo[lst,n]],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, May 30 2009 *)
Select[Range[200],PrimeQ[#^4+(#+1)^4]&] (* Harvey P. Dale, Jun 15 2013 *)
Position[Total/@Partition[Range[200]^4,2,1],?PrimeQ]//Flatten (* _Harvey P. Dale, Sep 20 2023 *)

A194155 Primes of the form k^8 + (k+1)^8.

Original entry on oeis.org

257, 2070241, 17995718017, 188386299457, 2505920246017, 3192202523137, 5072985298081, 11905609260481, 21370852274017, 766108283826337, 970961614082017, 2348771079002657, 2887223180589697, 9007197376151521, 55110306149736577, 77802445498340417

Offset: 1

Jonathan Vos Post, Aug 17 2011

Prime 8-dimensional centered cube numbers. This is to dimension 8 as A152913 is to dimension 4.

			a(2) = 5^8 + (5+1)^8 = 2070241 is prime.
a(3) = 17^8 + (17+1)^8.
a(4) = 23^8 + (23+1)^8.
a(5) = 32^8 + (32+1)^8.
a(6) = 33^8 + (33+1)^8.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A000040, A153504, A152913.

[ a: n in [0..200] | IsPrime(a) where a is n^8+(n+1)^8 ];  // Vincenzo Librandi, Dec 07 2011

Select[Table[n^8+(n+1)^8,{n,0,900}],PrimeQ] (* Vincenzo Librandi, Dec 07 2011 *)

A078902 Generalized Fermat primes of the form (k+1)^2^m + k^2^m, with m>1.

Original entry on oeis.org

17, 97, 257, 337, 881, 3697, 10657, 16561, 49297, 65537, 66977, 89041, 149057, 847601, 988417, 1146097, 1972097, 2070241, 2522257, 2836961, 3553777, 3959297, 4398577, 5385761, 7166897, 11073217, 17653681, 32530177, 41532497, 44048497

Offset: 1

T. D. Noe, Dec 12 2002

nonn

For k=1, these are the Fermat primes A019434. Is the set of generalized Fermat primes infinite? Conjecture that there are only a finite number of generalized Fermat primes for each value of k. See A077659, which shows that in cases such as k=11, there appear to be no primes. See A078901 for generalized Fermat numbers.

See A080131 for the conjectured number of primes for each k. See A080208 for the least k such that (k+1)^2^n + k^2^n is prime. The largest probable prime of this form discovered to date is the 10217-digit 312^2^12 + 311^2^12.

T. D. Noe, Table of n, a(n) for n = 1..525 (terms < 10^14)
T. D. Noe, Table of generalized Fermat primes of the form (k+1)^2^m + k^2^m
Eric Weisstein's World of Mathematics, Generalized Fermat Number

Cf. A019434, A077659, A078900, A078901.

Cf. A080131, A080208, A019434, A078902, A080134, A153504, A152913, A194185.

lst3=Select[lst2, PrimeQ[ # ]&] (* lst2 is from A078901 *)

A174156 Numbers n such that n^32+(n+1)^32 is a prime.

Original entry on oeis.org

8, 10, 12, 22, 100, 146, 154, 219, 246, 269, 287, 309, 336, 373, 392, 398, 423, 440, 449, 476, 515, 540, 557, 628, 671, 693, 715, 733, 746, 780, 848, 879, 913, 924, 926, 937, 974, 975, 1130, 1191, 1193, 1198, 1204, 1260, 1272, 1316, 1378, 1400, 1414, 1451

Offset: 1

Vincenzo Librandi, Mar 10 2010

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

Cf. A153504, A154535, A155211.

[n: n in [1..1500] |  IsPrime(n^32+(n+1)^32 )];

lst={};Do[If[PrimeQ[n^32+(n+1)^32],AppendTo[lst,n]],{n,1500}];lst (* Vincenzo Librandi, Aug 31 2012 *)
Select[Range[1500], PrimeQ[#^32 + (# + 1)^32] &] (* Bruno Berselli, Aug 31 2012 *)
Position[Total/@Partition[Range[1460]^32,2,1],?(PrimeQ[#]&)]//Flatten (* _Harvey P. Dale, Jul 05 2021 *)

A174157 Numbers n such that n^64+(n+1)^64 is a prime.

Original entry on oeis.org

95, 302, 443, 546, 755, 850, 878, 962, 983, 988, 1014, 1026, 1349, 1433, 1541, 1711, 1735, 1897, 1901, 1958, 1961, 1966, 2052, 2058, 2070, 2096, 2142, 2167, 2170, 2208, 2333, 2421, 2471, 2490, 2503, 2527, 2571, 2637, 2643, 2813, 2820, 2885, 2994

Offset: 1

Vincenzo Librandi, Mar 10 2010

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A153504, A154535, A155211.

[n: n in [0..2000] | IsPrime(n^64+(n+1)^64)]

lst={}; Do[If[PrimeQ[n^64+(n+1)^64], AppendTo[lst, n]], {n, 3000}]; lst (* Vincenzo Librandi_, Aug 31 2012 *)
Position[Total/@Partition[Range[3000]^64,2,1],?(PrimeQ[#]&)]//Flatten (* _Harvey P. Dale, Aug 01 2021 *)

A215431 Numbers n such that n^128+(n+1)^128 is a prime.

Original entry on oeis.org

31, 37, 65, 191, 255, 287, 359, 786, 836, 1178, 1229, 1503, 1601, 1609, 2093, 2103, 2254, 2307, 2471, 2934, 2978, 3215, 3220, 3363, 3402, 3705, 3724, 3892, 3894, 3976, 4094, 4478, 4490, 4535, 4566, 4683, 4749, 4752, 4789, 4918, 5064, 6061, 6162, 6167

Offset: 1

Vincenzo Librandi, Aug 31 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..250

Cf. A027861, A153504, A154535, A155211, A174156, A174157, A215432.

Select[Range[7000], PrimeQ[#^128 + (# + 1)^128] &]

A215432 Numbers n such that n^256+(n+1)^256 is a prime.

Original entry on oeis.org

85, 86, 157, 190, 195, 421, 504, 539, 621, 895, 1018, 1159, 1314, 1463, 1482, 1538, 1959, 2036, 2368, 2537, 2618, 2651, 3085, 3148, 3205, 3230, 3347, 3370, 3807, 4061, 4089, 4448, 4641, 4697, 4723, 4851, 4945

Offset: 1

Vincenzo Librandi, Aug 31 2012

Cf. A027861, A153504, A154535, A155211, A174156, A174157, A215431.

Select[Range[5000], PrimeQ[#^256 + (# + 1)^256] &];

A080208 a(n) is the least k such that the generalized Fermat number (k+1)^(2^n) + k^(2^n) is prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 8, 95, 31, 85, 59, 1078, 754, 311, 3508, 1828, 49957, 22844

Offset: 0

T. D. Noe, Feb 10 2003

The first five terms correspond to the five known Fermat primes. The sequence A078902 lists some of the generalized Fermat primes. Bjorn and Riesel examined generalized Fermat numbers for k <= 11 and n <= 999. The sequence A080134 lists the conjectured number of primes for each k.

For n >= 10, a(n) yields a probable prime. a(13) was found by Henri Lifchitz. It is known that a(14) > 1000.

			a(5) = 8 because (k+1)^32 + k^32 is prime for k = 8 and composite for k < 8.

T. D. Noe, Table of generalized Fermat primes of the form (k+1)^2^m + k^2^m
Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), no. 221, pp. 441-446.
Eric Weisstein's World of Mathematics, Generalized Fermat Number

Cf. A019434, A078902, A080134, A153504, A152913, A194185, A253633.

a(n) = A253633(n) - 1.

a(14)-a(15) from Jeppe Stig Nielsen, Nov 27 2020

a(16) by Kellen Shenton communicated by Jeppe Stig Nielsen, May 19 2023

A215433 Numbers n such that n^512 + (n+1)^512 is a prime.

Original entry on oeis.org

59, 864, 1455, 1723, 2118, 2172, 2460, 2851, 2916, 2971, 3193, 3476, 3747, 3782, 3795

Offset: 1

Vincenzo Librandi, Aug 31 2012

Cf. A027861, A153504, A154535, A155211, A174156, A174157, A215431, A215432.

Select[Range[0, 4000], PrimeQ[#^512 + (# + 1)^512] &]
Position[Partition[Range[3800]^512,2,1],?(PrimeQ[Total[#]]&)]//Flatten (* _Harvey P. Dale, Aug 03 2024 *)

A274234 Numbers n such that n^1024 + (n+1)^1024 is prime.

Original entry on oeis.org

1078, 2020, 2471, 3255, 4200, 5135, 5185, 6218, 6823, 7220, 8416, 9003, 9008, 9267, 9396, 9689, 10316, 11150, 11250, 11543, 11652, 12960, 14021, 14201, 16523, 16751, 17006, 17054, 17747, 17874, 18157, 18640, 18834, 20478, 20481, 20794, 21147, 22166, 22608, 22638, 24450, 24677, 24894, 25709

Offset: 1

Tim Johannes Ohrtmann, Jun 15 2016

nonn

The first five terms are certified primes, according to: factordb/certoverview.php. The others are probable primes. - Lewis Baxter, Jan 05 2021

Richard Fischer, Primzahlen mit der Form (B+1)^N+B^N.
Henri Lifchitz & Renaud Lifchitz, Search for : x^1024+y^1024.

Cf. A005097, A027861, A155211, A153504, A154535, A174156, A174157, A215431, A215432, A215433, A274235, A274236, A274237.

[n: n in [1..10000] |IsPrime(n^1024 + (n+1)^1024)]

Select[Range[1, 10000], PrimeQ[#^1024 + (#+1)^1024] &]

for(n=1, 10000, if(isprime(n^1024 + (n+1)^1024), print1(n, ", ")))

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A155211 Numbers n such that n^4+(n+1)^4 is a prime.

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A194155 Primes of the form k^8 + (k+1)^8.

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A078902 Generalized Fermat primes of the form (k+1)^2^m + k^2^m, with m>1.

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A174156 Numbers n such that n^32+(n+1)^32 is a prime.

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A174157 Numbers n such that n^64+(n+1)^64 is a prime.

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A215431 Numbers n such that n^128+(n+1)^128 is a prime.

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A215432 Numbers n such that n^256+(n+1)^256 is a prime.

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A080208 a(n) is the least k such that the generalized Fermat number (k+1)^(2^n) + k^(2^n) is prime.

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A215433 Numbers n such that n^512 + (n+1)^512 is a prime.

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