A089001 -id:A089001 - cp's OEIS frontend

A236767 Numbers whose square is a fourth power plus a prime.

2, 10, 37, 82, 442, 577, 730, 901, 1090, 1297, 1765, 2026, 4357, 5185, 5626, 7570, 8650, 9217, 9802, 10405, 11026, 15130, 17425, 18226, 23410, 24337, 26245, 31330, 34597, 35722, 40402, 41617, 47962

Offset: 1

Hans Havermann, Jan 30 2014

nonn

Based on a 1999 observation of Alessandro Zaccagnini (via John Robertson) intended to dissuade expectation of a finite fourth-power analogy to A020495, A045911.

It can be shown that A089001^2 + 1 are members of this sequence. David Applegate shows that they are the only members: If x^2 = y^4 + p, let a = x - y^2. Then y^4 + p = x^2 = (y^2 + a)^2 = y^4 + 2a*y^2 + a^2, so p = 2a*y^2 + a^2, and so a divides p. Since p is a prime, a must be a unit (that is, +1 or -1). But since p >= 2, a must be +1.

			2 is a term because 2^2 = 1^4 + 3;
10 is a term because 10^2 = 3^4 + 19;
37 is a term because 37^2 = 6^4 + 73.

Hans Havermann, Table of n, a(n) for n = 1..1000
John Robertson, Integers of the form x^2+kp (see last paragraph)

Cf. A020495, A045911, A089001.

r=Range[10000]^4; j=1; Do[c=i^2; k=c^2-Take[r,i]; Do[c++; j=j+2; k=k+j; If[MemberQ[PrimeQ[k], True], Print[c]], {2*i+1}], {i, 10000}] (* brute force *)
s=A089001; s^2+1 (* based on formula *)

A089001^2 + 1

A239920 Integers n such that 2n^2+1, 2n^3+1 and 2n^4+1 are prime.

Original entry on oeis.org

1, 6, 21, 30, 96, 297, 375, 621, 1359, 1704, 1749, 1761, 3696, 3849, 4467, 8025, 8646, 9834, 11352, 15630, 17397, 17949, 19575, 20274, 27087, 28452, 30504, 32154, 32307, 33666, 35670, 36240, 37785, 37962, 39927, 40617, 42987, 44250, 47559, 49335, 49599

Offset: 1

Zak Seidov, Mar 29 2014

nonn

Zak Seidov, Table of n, a(n) for n = 1..1000

Subsequence of A239874. Cf. A089001, A168550.

Select[Range[0, 50000], PrimeQ[2 #^2 + 1] && PrimeQ[2  #^3 + 1] && PrimeQ[2 #^4 + 1]&] (* Vincenzo Librandi, Mar 30 2014 *)

s=[]; for(n=1, 100000, if(isprime(2*n^2+1) && isprime(2*n^3+1) && isprime(2*n^4+1), s=concat(s, n))); s \\ Colin Barker, Mar 29 2014

A240099 Integers n such that 2n^k + 1, for k = 2..6, are prime.

Original entry on oeis.org

1, 44250, 1004004, 3490575, 3517335, 5750115, 10729026, 19193559, 20251770, 25284039, 25552194, 30204801, 33733206, 39015405, 47518809, 52463445, 58370025, 69502971, 72009429, 77086380, 78510156, 83972646, 85955475, 89190969, 90499584, 92246199, 95374005

Offset: 1

Zak Seidov, Apr 01 2014

nonn

Note that 2n^7+1 may or may not be prime.
First n>1 such that 2n^k+1, for k=2..7, are prime, is a(4) = 3490575.
First n>1 such that 2n^k+1, for k=2..8, are prime, is 83972646.
Subsequence of A239925:  a(2) = 44250 = A239925(4),  a(3) = 1004004 = A239925(27).

Cf. A089001, A168550, A239874, A239925.

Select[Range[10^8],AllTrue[2#^Range[2,6]+1,PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 18 2015 *)

A115272 Primes p such that p + 2, 18p^2 + 1, and 18(p+2)^2 + 1 are all primes.

Original entry on oeis.org

29, 107, 431, 1487, 1607, 2141, 5501, 10139, 10271, 17579, 22481, 23057, 27479, 32369, 36341, 36929, 38447, 55931, 57527, 69827, 75539, 78539, 79691, 81047, 81971, 84179, 86027, 89561, 93761, 102059, 112571, 113147, 118799, 119687

Offset: 1

Zak Seidov, Jan 19 2006

nonn

			a(1)=29 because 31, 18*29^2 + 1 = 15139, and 18*31^2 + 1 = 17299 are all primes.

Cf. A089001 (Numbers n such that 2*n^2 + 1 is prime),
A090612 (Numbers k such that the k-th prime is of the form 2*k^2+1),
A090698 (Primes of the form 2*n^2+1),
A113541 (Numbers n such that 18*n^2+1 is a multiple of 19).

[p: p in PrimesUpTo(200000)| IsPrime(p+2) and IsPrime(18*p^2+1) and IsPrime(18*(p+2)^2+1)] // Vincenzo Librandi, Nov 13 2010

More terms from Vincenzo Librandi, Mar 27 2010

A117132 Numbers n such that 5*n^5 + 1 is prime.

Original entry on oeis.org

8, 14, 32, 80, 138, 144, 236, 284, 302, 342, 344, 390, 420, 438, 446, 510, 542, 546, 570, 612, 638, 644, 680, 690, 696, 768, 794, 804, 812, 816, 822, 834, 866, 888, 908, 942, 960, 1020, 1022, 1148, 1190, 1194, 1224, 1250, 1278, 1358, 1368, 1398, 1434, 1446

Offset: 1

Parthasarathy Nambi, Apr 20 2006

			If n=144 then 5*n^5 + 1 = 309586821121 (prime).

Cf. A089001.

Select[Range[2000], PrimeQ[5#^5 + 1] &] (* Stefan Steinerberger and Giovanni Resta, Apr 22 2006 *)

is(n)=isprime(5*n^5+1) \\ Charles R Greathouse IV, Jun 13 2017

More terms from Stefan Steinerberger and Giovanni Resta, Apr 22 2006

A173416 Exactly one of 2n^2-1 and 2n^2+1 is prime.

Original entry on oeis.org

1, 2, 4, 7, 8, 9, 10, 11, 13, 15, 17, 18, 22, 25, 27, 28, 30, 33, 34, 38, 39, 41, 43, 46, 49, 50, 52, 56, 59, 62, 63, 64, 66, 69, 72, 73, 75, 76, 80, 81, 85, 91, 92, 93, 95, 96, 98, 99, 105, 108, 109, 112, 113, 115, 118, 123, 125, 126, 127, 134, 135, 137, 140, 141, 143

Offset: 1

Juri-Stepan Gerasimov, Mar 01 2010

nonn

Numbers in A089001 or in A066049 but not in both. [From R. J. Mathar, Mar 09 2010]

			a(1)=1 because 2*1^2-1=1 is nonprime and 2*1^2+1=3 is prime.

37 removed by R. J. Mathar, Mar 09 2010

A240105 Integers m such that 2*m^k + 1, for k = 2..7, are prime.

Original entry on oeis.org

1, 3490575, 83972646, 414180489, 476072025, 1881147720, 3020243916, 3188924769, 3285167214, 3543143220, 6593858205, 8239349955, 10914074124, 14102235060, 15455042889, 16196415300, 16588528539, 16636093485, 17688635511, 17929182270, 18997337436, 19290317670, 19347263739

Offset: 1

Zak Seidov, Apr 01 2014

nonn

First m>1 such that 2*m^k+1, for k=2..8, are prime, is a(3) = 83972646.

Subsequence of A240099: a(2) = 3490575 = A240099(4), a(3) = 83972646 = A240099(22).

Cf. A089001, A168550, A239874, A239925, A240099.

More terms from Jinyuan Wang, Jun 12 2025

cp's OEIS Frontend

A236767 Numbers whose square is a fourth power plus a prime.

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A239920 Integers n such that 2n^2+1, 2n^3+1 and 2n^4+1 are prime.

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A240099 Integers n such that 2n^k + 1, for k = 2..6, are prime.

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A115272 Primes p such that p + 2, 18*p^2 + 1, and 18*(p+2)^2 + 1 are all primes.

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Magma

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A117132 Numbers n such that 5*n^5 + 1 is prime.

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Mathematica

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A173416 Exactly one of 2n^2-1 and 2n^2+1 is prime.

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A240105 Integers m such that 2*m^k + 1, for k = 2..7, are prime.

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Extensions

A115272 Primes p such that p + 2, 18p^2 + 1, and 18(p+2)^2 + 1 are all primes.