A005574 -id:A005574 - cp's OEIS frontend

A246452 Let A be the set of all numbers m for which m^2+1 is prime (see A005574). The sequence lists the pairs of numbers (a, b) such that every m > 1 included in A can be written in the form m = a + b for a <= b both in A and a minimum.

1, 1, 2, 2, 2, 4, 4, 6, 4, 10, 2, 14, 4, 16, 4, 20, 2, 24, 10, 26, 4, 36, 14, 40, 2, 54, 10, 56, 20, 54, 10, 74, 6, 84, 4, 90, 16, 94, 6, 110, 4, 116, 4, 120, 2, 124, 4, 126, 4, 130, 16, 130, 4, 146, 6, 150, 4, 156, 10, 160, 6, 170, 4, 176, 4, 180, 20, 184

Offset: 1

Michel Lagneau, Aug 26 2014

nonn

Conjecture: for n>1, A005574(n) = a+b where a and b are integers in A005574.

By definition, a(2n-1) + a(2n) = A005574(n+1), and a(2n-1) is the minimal solution where the pair (a(2n-1), a(2n)) are both in A005574. The above conjecture says such a pair always exists. - Jens Kruse Andersen, Aug 27 2014

			The 3rd pair (2,4) is in the sequence because for m = 2,4 and 2+4 = 6 we obtain three primes 2^2+1 = 5, 4^2+1 = 17 and 6^2+1 = 37.
The 6th number above 1 in A005574 is 16 which can be written as 2+14 or 6+10 using a pair in A005574. 2 is the minimal number in a solution so the 6th pair is (a(11), a(12)) = (2, 14). - _Jens Kruse Andersen_, Aug 27 2014

Michel Lagneau, Table of n, a(n) for n = 1..5000

Cf. A002496, A005574.

with(numtheory):lst:={}:nn:=10000:
for i from 1 to nn do:
  if type(i^2+1,prime)=true
  then
  lst:=lst union {i}:
  else
  fi:
od:
n1:=nops(lst):
  for n from 1 to 50 do:
  m:=lst[n]:ii:=0:
    for u from 1 to n1 while(ii=0)do:
      for v from u to n1 while(ii=0)do:
        a:=lst[u]:b:=lst[v]:
        if m=a+b
        then
        ii:=1: printf(`%d, `,a): printf(`%d, `,b):
        else
        fi:
      od:
    od:
   od:

A277970 Numbers k = A005574(m) such that k = (A005574(m-1)+A005574(m+1))/2.

Original entry on oeis.org

4, 10, 20, 120, 130, 180, 230, 260, 440, 470, 680, 700, 750, 920, 1060, 1320, 1736, 1860, 1970, 2106, 2320, 2460, 2760, 2850, 2890, 3074, 3660, 3800, 4180, 4370, 5030, 5236, 5304, 5814, 5990, 6130, 6350, 6590, 6724, 6780, 6990, 7190, 7384, 7520, 7744, 8180

Offset: 1

Michel Lagneau, Nov 07 2016

nonn

Subsequence of A005574.
a(n) == 0, 4 or 6 (mod 10).
The corresponding primes a(n)^2+1 are 17, 101, 401, 14401, 16901, ...
The numbers j are symmetric centers given by the middle j of each triple of integers (i, j, k) = (2, 4, 6), (6, 10, 14), (16, 20, 24), (116, 120, 124), ... which are elements of A005574. This symmetry can be seen from the differences between the numbers of each triple. From these, we obtain the following differences (2, 2), (4, 4), (4, 4), (4, 4), .... More generally, a symmetric center may also be the middle of a m-tuple of m even integers (i(1), i(2), ..., i(m)) with m odd, where i(1)^2+1, i(2)^2+1, ..., i(m)^2+1 are m consecutive primes. In order to obtain the symmetry, there must be (i(1)+i(m))/2 = (i(2)+i(m-1))/2 = ... = (i((m-1)/2)+i((m+3)/2))/2 = i((m+1)/2), the middle of the m-tuple.
Because an m-tuple is not unique for each a(n), we introduce the notion of order O(a(n)) (see the table below). The calculations show that O(a(n)) < = 4 for n < 500000.
+------+-----+--------------------------------------+-------------------+
| a(n) |order|               m-tuples               |    differences    |
+------+-----+--------------------------------------+-------------------+
|    4 |  1  | (2,4,6)                              |(2, 2)             |
|   10 |  2  | (6,10,14)                            |(4, 4)             |
|      |     | (4,6,10,14,16)                       |(2,4,4,2)          |
|   20 |  1  | (16,20,24)                           |(4,4)              |
|  120 |  1  | (116,120,124)                        |(4,4)              |
|  130 |  1  | (126,130,134)                        |(4,4)              |
|  180 |  1  | (176,180,184)                        |(4,4)              |
|  230 |  1  | (224,230,236)                        |(6,6)              |
|  260 |  4  | (256,260,264)                        |(4,4)              |
|      |     | (250,256,260,264,270)                |(6,4,4,6)          |
|      |     | (240,250,256,260,264,270,280)        |(10,6,4,4,6,10)    |
|      |     | (236,240,250,256,260,264,270,280,284)|(4,10,6,4,4,4,10,4)|
|  440 |  1  | (436,440,444)                        |(4, 4)             |
...
Former name was:
Numbers j = (i + k)/2 such that i^2+1, j^2+1 and k^2+1 are three consecutive primes.- Robert Israel, Jun 19 2019

			10 is in the sequence because from the triple (i, j, k) = (6, 10, 14) with j = (i + k)/2 = (6+14)/2 = 10, we obtain the three consecutive primes (i^2+1, j^2+1, k^2+1) = (37, 101, 197).

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

Cf. A002496, A005574, A006562.

nn:=100000:V:=array(1..6656):kk:=0:
for i from 1 to nn do:
x:=i^2+1:
  if isprime(x)
   then
   kk:=kk+1:V[kk]:=i:
  else
fi:
od:
for n from 2 to kk-2 do:
   p:=V[n]:m:=V[n+1]:q:=V[n+2]:
    if (p+q)/2 = m
     then
      ii:=1:printf(`%d, `,V[n+1]):
    else
   fi:
od:

P = Select[Range[10^4]^2+1, PrimeQ]; Reap[Do[{i, j, k} = Sqrt[P[[n ;; n+2]] - 1]; If[AllTrue[{i, j, k}, IntegerQ] && (i+k)/2 == j, Print[{i, j, k}]; Sow[j]], {n, 1, Length[P]-2}]][[2, 1]] (* Jean-François Alcover, Nov 08 2016 *)

Name changed by Robert Israel, Jun 19 2019

A259940 Let A={A005574(n)}, the set of all numbers n for which n^2+1 is prime. The sequence lists the number of decompositions A005574(n) = A005574(n1) + A005574(n2) for some n1, n2 and every A005574(n)>1.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 3, 2, 3, 2, 4, 1, 3, 2, 1, 1, 4, 4, 5, 2, 5, 3, 5, 8, 5, 5, 8, 6, 7, 7, 6, 7, 6, 6, 5, 8, 7, 8, 7, 11, 12, 6, 12, 8, 11, 12, 8, 11, 9, 8, 10, 13, 11, 6, 10, 8, 12, 11, 13, 12, 10, 17, 9, 8, 10, 13, 11, 15, 11, 9, 8, 14, 13, 12, 8, 8, 7, 9, 7

Offset: 1

Michel Lagneau, Jul 09 2015

nonn

We use a little-known conjecture by Goldbach on the primes of form n^2+1: let A be the set of all numbers a for which a^2+1 is prime (A={1, 2, 4, 6, 10, ...}). Then every a in A (a>1) can be written in the form a=b+c for b,c in A.

			a(20)=5 because A005574(20)= 110 =>
A005574(20)= A005574(7)  + A005574(19)= 16 + 94,
A005574(20)= A005574(8)  + A005574(18)= 20 + 90,
A005574(20)= A005574(10) + A005574(17)= 26 + 84,
A005574(20)= A005574(11) + A005574(16)= 36 + 74,
A005574(20)= A005574(13) + A005574(14)= 54 + 56, for a total of five decompositions.

Mathoverflow, Primes of the form a^2+1

Cf. A005574.

T:=array(1..112):
nn:=1000:k:=0:
for i from 1 to nn do:
p:=i^2+1:if type(p,prime)=true
then
k:=k+1:T[k]:=i:
else fi:
od:
for n from 1 to k do:q:=T[n]:it:=0:
  for a from 1 to k do:p1:=T[a]:
   for b from a to k do:p2:=T[b]:
    if q=p1+p2
    then
    it:=it+1:
    else fi:
   od:
od:
printf(`%d, `,it):
od:

A002496 Primes of the form k^2 + 1.

Original entry on oeis.org

2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601, 2917, 3137, 4357, 5477, 7057, 8101, 8837, 12101, 13457, 14401, 15377, 15877, 16901, 17957, 21317, 22501, 24337, 25601, 28901, 30977, 32401, 33857, 41617, 42437, 44101, 50177

Offset: 1

N. J. A. Sloane

It is conjectured that this sequence is infinite, but this has never been proved.
An equivalent description: primes of form P = (p1*p2*...*pm)^k + 1 where p1..pm are primes and k > 1, since then k must be even for P to be prime.
Also prime = p(n) if A054269(n) = 1, i.e., quotient-cycle-length = 1 in continued fraction expansion of sqrt(p). - Labos Elemer, Feb 21 2001
Also primes p such that phi(p) is a square.
Also primes of form x*y + z, where x, y and z are three successive numbers. - Giovanni Teofilatto, Jun 05 2004
It is a result that goes back to Mirsky that the set of primes p for which p-1 is squarefree has density A, where A = A005596 denotes the Artin constant. More precisely, Sum_{p <= x} mu(p-1)^2 = A*x/log x + o(x/log x) as x tends to infinity. Conjecture: Sum_{p <= x, mu(p-1)=1} 1 = (A/2)*x/log x + o(x/log x) and Sum_{p <= x, mu(p-1)=-1} 1 = (A/2)*x/log x + o(x/log x). - Pieter Moree (moree(AT)mpim-bonn.mpg.de), Nov 03 2003
Also primes of the form x^y + 1, where x > 0, y > 1. Primes of the form x^y - 1 (x > 0, y > 1) are the Mersenne primes listed in A000668(n) = {3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ...}. - Alexander Adamchuk, Mar 04 2007
With the exception of the first two terms {2,5}, the continued fraction (1 + sqrt(p))/2 has period 3. - Artur Jasinski, Feb 03 2010
With the exception of the first term {2}, congruent to 1 (mod 4). - Artur Jasinski, Mar 22 2011
With the exception of the first two terms, congruent to 1 or 17 (mod 20). - Robert Israel, Oct 14 2014
From Bernard Schott, Mar 22 2019: (Start)
These primes are the primitive terms which generate the sequence of integers with only one prime factor and whose Euler's totient is a square: A054755. So this sequence is a subsequence of A054755 and of A039770. Additionally, the terms of this sequence also have a square cototient, so this sequence is a subsequence of A063752 and A054754.
If p prime = n^2 + 1, phi(p) = n^2 and cototient(p) = 1^2.
Except for 3, the four Fermat primes in A019434 {5, 17, 257, 65537}, belong to this sequence; with F_k = 2^(2^k) + 1, phi(F_k) = (2^(2^(k-1)))^2.
See the file "Subfamilies and subsequences" (& I) in A039770 for more details, proofs with data, comments, formulas and examples. (End)
In this sequence, primes ending with 7 seem to appear twice as often as primes ending with 1. This is because those with 7 come from integers ending with 4 or 6, while those with 1 come only from integers ending with 0 (see De Koninck & Mercier reference). - Bernard Schott, Nov 29 2020
The set of odd primes p for which every elliptic curve of the form y^2 = x^3 + d*x has order p-1 over GF(p) for those d with (d,p)=1 and d a fourth power modulo p. - Gary Walsh, Sep 01 2021 [edited, Gary Walsh, Apr 26 2025]

Jean-Marie De Koninck & Armel Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 211 pp. 34 and 169, Ellipses, Paris, 2004.
Leonhard Euler, De numeris primis valde magnis (E283), reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 3, p. 22.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 17.
Hugh L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 208.
C. Stanley Ogilvy, Tomorrow's Math. 2nd ed., Oxford Univ. Press, 1972, p. 116.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 118.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Rev. ed. 1997), p. 134.

T. D. Noe, Table of n, a(n) for n = 1..10000
Tewodros Amdeberhan, Luis A. Medina and Victor H. Moll, Arithmetical properties of a sequence arising from an arctangent sum, J. Numb. Theory, Vol. 128, No. 6 (2008), pp. 1807-1846, eq. (1.10).
William D. Banks, John B. Friedlander, Carl Pomerance and Igor E. Shparlinski, Multiplicative structure of values of the Euler function, in High Primes and Misdemeanours: Lectures in Honour of the Sixtieth Birthday of Hugh Cowie Williams (A. Van der Poorten, ed.), Fields Inst. Comm. 41 (2004), pp. 29-47.
Paul T. Bateman and Roger A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Mathematics of Computation, Vol. 16, No. 79 (1962), pp. 363-367.
Joel E. Cohen, Conjectures about Primes and Cyclic Numbers, arXiv:2508.08335 [math.NT], 2025. See p. 11.
Frank Ellermann, Primes of the form (m^2)+1 up to 10^6.
Dan Ismailescu and Yunkyu James Lee, Polynomially growing integer sequences all whose terms are composite, arXiv:2501.04851 [math.NT], 2025. See p. 9.
Leon Mirsky, The number of representations of an integer as the sum of a prime and a k-free integer, Amer. Math. Monthly, Vol. 56, No. 1 (1949), pp. 17-19.
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
Apoloniusz Tyszka, On sets X subset of N for which we know an algorithm that computes a threshold number t(X) in N such that X is infinite if and only if X contains an element greater than t(X), 2019.
Apoloniusz Tyszka and Sławomir Kurpaska, Open problems that concern computable sets X, subset of N, and cannot be formally stated as they refer to current knowledge about X, (2020).
Eric Weisstein's World of Mathematics, Landau's Problems.
Eric Weisstein's World of Mathematics, Near-Square Prime.
Wikipedia, Bateman-Horn Conjecture.
Marek Wolf, Search for primes of the form m^2+1, arXiv:0803.1456 [math.NT], 2008-2010.

Cf. A083844 (number of these primes < 10^n), A199401 (growth constant).
Cf. A001912, A005574, A054964, A062325, A088179, A090693, A141293, A172168.
Cf. A000668 (Mersenne primes), A019434 (Fermat primes).
Subsequence of A039770.
Cf. A010051, subsequence of A002522.
Cf. A237040 (an analog for n^3 + 1).
Cf. A010051, A000290; subsequence of A028916.
Subsequence of A039770, A054754, A054755, A063752.
Primes of form n^2+b^4, b fixed: A243451 (b=2), A256775 (b=3), A256776 (b=4), A256777 (b=5), A256834 (b=6), A256835 (b=7), A256836 (b=8), A256837 (b=9), A256838 (b=10), A256839 (b=11), A256840 (b=12), A256841 (b=13).
Cf. A030430 (primes ending with 1), A030432 (primes ending with 7).

a002496 n = a002496_list !! (n-1)
a002496_list = filter ((== 1) . a010051') a002522_list
-- Reinhard Zumkeller, May 06 2013

[p: p in PrimesUpTo(100000)| IsSquare(p-1)]; // Vincenzo Librandi, Apr 09 2011

select(isprime, [2, seq(4*i^2+1, i= 1..1000)]); # Robert Israel, Oct 14 2014

Select[Range[100]^2+1, PrimeQ]
Join[{2},Select[Range[2,300,2]^2+1,PrimeQ]] (* Harvey P. Dale, Dec 18 2018 *)

isA002496(n) = isprime(n) && issquare(n-1) \\ Michael B. Porter, Mar 21 2010

is_A002496(n)=issquare(n-1)&&isprime(n) \\ For "random" numbers in the range 10^10 and beyond, at least 5 times faster than the above. - M. F. Hasler, Oct 14 2014

# Python 3.2 or higher required
from itertools import accumulate
from sympy import isprime
A002496_list = [n+1 for n in accumulate(range(10**5),lambda x,y:x+2*y-1) if isprime(n+1)] # Chai Wah Wu, Sep 23 2014

# Python 2.4 or higher required
from sympy import isprime
A002496_list = list(filter(isprime, (n*n+1 for n in range(10**5)))) # David Radcliffe, Jun 26 2016

There are O(sqrt(n)/log(n)) terms of this sequence up to n. But this is just an upper bound. See the Bateman-Horn or Wolf papers, for example, for the conjectured for what is believed to be the correct density.
a(n) = 1 + A005574(n)^2. - R. J. Mathar, Jul 31 2015
Sum_{n>=1} 1/a(n) = A172168. - Amiram Eldar, Nov 14 2020
a(n+1) = 4*A001912(n)^2 + 1. - Hal M. Switkay, Apr 03 2022

Formula, reference, and comment from Charles R Greathouse IV, Aug 24 2009

Edited by M. F. Hasler, Oct 14 2014

A000068 Numbers k such that k^4 + 1 is prime.

Original entry on oeis.org

1, 2, 4, 6, 16, 20, 24, 28, 34, 46, 48, 54, 56, 74, 80, 82, 88, 90, 106, 118, 132, 140, 142, 154, 160, 164, 174, 180, 194, 198, 204, 210, 220, 228, 238, 242, 248, 254, 266, 272, 276, 278, 288, 296, 312, 320, 328, 334, 340, 352, 364, 374, 414, 430, 436, 442, 466

Offset: 1

N. J. A. Sloane

Harvey Dubner, Generalized Fermat primes, J. Recreational Math., 18 (1985): 279-280.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
M. Lal, Primes of the form n^4 + 1, Math. Comp., 21 (1967), 245-247.
D. Shanks, On numbers of the form n^4+1, Math. Comp. 15 (74) (1961), 186-189.

Cf. A002523, A037896, A005574, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.

[n: n in [0..800] | IsPrime(n^4+1)]; // Vincenzo Librandi, Nov 18 2010

Select[Range[10^2*2], PrimeQ[ #^4+1] &] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)

{a(n) = local(m); if( n<1, 0, for(k=1, n, until( isprime(m^4 + 1), m++)); m)};

list(lim)=my(v=List([1])); forstep(k=2,lim,2, if(isprime(k^4+1), listput(v,k))); Vec(v) \\ Charles R Greathouse IV, Mar 31 2022

1+a(n)^4 = A037896(n).

A067201 Numbers k such that k^2 + 2 is prime.

Original entry on oeis.org

0, 1, 3, 9, 15, 21, 33, 39, 45, 57, 81, 99, 105, 111, 117, 123, 147, 171, 219, 225, 237, 243, 249, 255, 273, 297, 303, 309, 321, 345, 351, 363, 369, 375, 387, 417, 423, 429, 441, 447, 453, 477, 501, 513, 549, 555, 561, 573, 603, 609, 651, 675, 681, 699, 711, 753

Offset: 1

Benoit Cloitre, Feb 19 2002

nonn

All terms > 1 are divisible by 3. - Robert Israel, Sep 05 2014

T. D. Noe, Table of n, a(n) for n=1..1000
Eric Weisstein's World of Mathematics, Near-Square Prime

Equals 6*A056900(n-2) + 3, n>1.
Cf. A106571, A040976.
Other sequences of the type "Numbers k such that k^2 + i is prime": A005574 (i=1), this sequence (i=2), A049422 (i=3), A007591 (i=4), A078402 (i=5), A114269 (i=6), A114270 (i=7), A114271 (i=8), A114272 (i=9), A114273 (i=10), A114274 (i=11), A114275 (i=12).

select(t -> isprime(t^2+2), [0,1,seq(3*i,i=1..1000)]); # Robert Israel, Sep 05 2014

lst={};Do[If[PrimeQ[n^2+2], AppendTo[lst, n]], {n, 3*10^2}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)
Join[{0, 1}, Select[Range[3, 1000, 6], PrimeQ[#^2 + 2] &]] (* Zak Seidov, Jan 30 2014 *)

select(n -> isprime(n^2+2),[1..500]) \\ Edward Jiang, Sep 05 2014

A001912 Numbers k such that 4*k^2 + 1 is prime.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 10, 12, 13, 18, 20, 27, 28, 33, 37, 42, 45, 47, 55, 58, 60, 62, 63, 65, 67, 73, 75, 78, 80, 85, 88, 90, 92, 102, 103, 105, 112, 115, 118, 120, 125, 128, 130, 132, 135, 140, 142, 150, 153, 157, 163, 170, 175, 192, 193, 198, 200

Offset: 1

N. J. A. Sloane

Complement of A094550. - Hermann Stamm-Wilbrandt, Sep 16 2014
Positive integers whose square is the sum of two triangular numbers in exactly one way (A000217(k) + A000217(k+1) = k*(k+1)/2 + (k+1)*(k+2)/2 = (k+1)^2). In other words, positive integers k such that A052343(k^2) = 1. - Altug Alkan, Jul 06 2016
4*a(n)^2 + 1 = A002496(n+1). - Hal M. Switkay, Apr 03 2022

E. Kogbetliantz and A. Krikorian, Handbook of First Complex Prime Numbers, Gordon and Breach, NY, 1971, p. 1.
M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 11.
C. S. Ogilvy, Tomorrow's Math. 2nd ed., Oxford Univ. Press, 1972, p. 116.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]
E. Kogbetliantz and A. Krikorian Handbook of First Complex Prime Numbers, Gordon and Breach, NY, 1971 [Annotated scans of a few pages]
Marek Wolf, Search for primes of the form m^2+1, arXiv:0803.1456 [math.NT], 2008-2010.

Cf. A002496, A005574, A062325, A090693, A094550, A214517 (first differences).

[n: n in [1..100] | IsPrime(4*n^2+1)] // Vincenzo Librandi, Nov 21 2010

A001912 := proc(n)
    option remember;
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isprime(4*a^2+1) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Aug 09 2012

Select[Range[200], PrimeQ[4#^2 + 1] &] (* Alonso del Arte, Dec 20 2013 *)

is(n)=isprime(4*n^2 + 1) \\ Charles R Greathouse IV, Apr 28 2015

a(n) = A005574(n+1)/2.

A006313 Numbers k such that k^16 + 1 is prime.

Original entry on oeis.org

1, 2, 44, 74, 76, 94, 156, 158, 176, 188, 198, 248, 288, 306, 318, 330, 348, 370, 382, 396, 452, 456, 470, 474, 476, 478, 560, 568, 598, 642, 686, 688, 690, 736, 774, 776, 778, 790, 830, 832, 834, 846, 900, 916, 946, 956, 972, 982, 984, 1018, 1044, 1078

Offset: 1

N. J. A. Sloane

nonn

Dubner, Harvey. "Generalized Fermat primes." J. Recreational Math., 18 (1985): 279-280.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Cf. A005574, A000068, A006314, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.

[ n: n in [0..1500] | IsPrime(n^16+1) ]; // Vincenzo Librandi, Nov 18 2010

lst={};Do[If[PrimeQ[n^16+1], AppendTo[lst, n]], {n, 10^4}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 20 2008 *)
s=Reap[Sow[1];Do[If[PrimeQ[n^16+1],Sow[n]],{n,2,40352,2}]][[2,1]] (* Zak Seidov, Dec 22 2010 *)
Join[{1},2*Flatten[Position[Range[2,1100,2]^16+1,?PrimeQ]]] (* _Harvey P. Dale, Jun 26 2017 *)

isA006313(n) = isprime(n^16+1) \\ Michael B. Porter, Mar 25 2010

More terms from Hugo Pfoertner, Jun 22 2003

A006314 Numbers k such that k^8 + 1 is prime.

Original entry on oeis.org

1, 2, 4, 118, 132, 140, 152, 208, 240, 242, 288, 290, 306, 378, 392, 426, 434, 442, 508, 510, 540, 542, 562, 596, 610, 664, 680, 682, 732, 782, 800, 808, 866, 876, 884, 892, 916, 918, 934, 956, 990, 1022, 1028, 1054, 1106, 1120, 1174, 1224, 1232, 1256, 1284

Offset: 1

N. J. A. Sloane

Dubner, Harvey. "Generalized Fermat primes." J. Recreational Math., 18 (1985): 279-280.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A005574, A000068, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.

[n: n in [0..2000] | IsPrime(n^8+1)]; // Vincenzo Librandi, Nov 18 2010

Select[Range[1300],PrimeQ[#^8+1]&]  (* Harvey P. Dale, Mar 31 2011 *)

isA006314(n) = isprime(n^8+1) \\ Michael B. Porter, Mar 24 2010

A258805(n) = a(n)^8+1. - R. J. Mathar, Jun 05 2025

More terms from James Sellers, May 03 2000

A246392 Numbers n such that Phi(10, n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

2, 3, 5, 10, 11, 12, 16, 20, 21, 22, 33, 37, 38, 43, 47, 48, 55, 71, 75, 76, 80, 81, 111, 121, 126, 131, 133, 135, 136, 141, 155, 157, 158, 165, 176, 177, 180, 203, 223, 242, 245, 251, 253, 256, 257, 258, 265, 268, 276, 286, 290, 297, 307, 322, 323, 342, 361, 363, 366, 375, 377, 385, 388, 396, 411

Offset: 1

Eric Chen, Nov 13 2014

nonn

Numbers n such that (n^5+1)/(n+1) is prime, or numbers n such that A060884(n) is prime.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 893 terms from Robert Price)
OEIS Wiki, Cyclotomic Polynomials at x=sigma(n)

Cf. A008864 (1), A006093 (2), A002384 (3), A005574 (4), A049409 (5), A055494 (6), A100330 (7), A000068 (8), A153439 (9), this sequence (10), A162862 (11), A246397 (12), A217070 (13), A006314 (16), A217071 (17), A164989 (18), A217072 (19), A217073 (23), A153440 (27), A217074 (29), A217075 (31), A006313 (32), A097475 (36), A217076 (37), A217077 (41), A217078 (43), A217079 (47), A217080 (53), A217081 (59), A217082 (61), A006315 (64), A217083 (67), A217084 (71), A217085 (73), A217086 (79), A153441 (81), A217087 (83), A217088 (89), A217089 (97), A006316 (128), A153442 (243), A056994 (256), A056995 (512), A057465 (1024), A057002 (2048), A088361 (4096), A088362 (8192), A226528 (16384), A226529 (32768), A226530 (65536).

Cf. A060884, A085398, A259257.

[n: n in [1..500]| IsPrime((n^5+1) div (n+1))]; // Vincenzo Librandi, Nov 14 2014

A246392:=n->`if`(isprime((n^5+1)/(n+1)),n,NULL): seq(A246392(n), n=1..500); # Wesley Ivan Hurt, Nov 15 2014

Select[Range[700], PrimeQ[(#^5 + 1) / (# + 1)] &] (* Vincenzo Librandi, Nov 14 2014 *)

for(n=1,10^3,if(isprime(polcyclo(10,n)),print1(n,", "))); \\ Joerg Arndt, Nov 13 2014

cp's OEIS Frontend

A246452 Let A be the set of all numbers m for which m^2+1 is prime (see A005574). The sequence lists the pairs of numbers (a, b) such that every m > 1 included in A can be written in the form m = a + b for a <= b both in A and a minimum.

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A277970 Numbers k = A005574(m) such that k = (A005574(m-1)+A005574(m+1))/2.

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A259940 Let A={A005574(n)}, the set of all numbers n for which n^2+1 is prime. The sequence lists the number of decompositions A005574(n) = A005574(n1) + A005574(n2) for some n1, n2 and every A005574(n)>1.

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A002496 Primes of the form k^2 + 1.

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A000068 Numbers k such that k^4 + 1 is prime.

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A067201 Numbers k such that k^2 + 2 is prime.

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A001912 Numbers k such that 4*k^2 + 1 is prime.

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