A000068 -id:A000068 - cp's OEIS frontend

A253240 Square array read by antidiagonals: T(m, n) = Phi_m(n), the m-th cyclotomic polynomial at x=n.

1, 1, -1, 1, 0, 1, 1, 1, 2, 1, 1, 2, 3, 3, 1, 1, 3, 4, 7, 2, 1, 1, 4, 5, 13, 5, 5, 1, 1, 5, 6, 21, 10, 31, 1, 1, 1, 6, 7, 31, 17, 121, 3, 7, 1, 1, 7, 8, 43, 26, 341, 7, 127, 2, 1, 1, 8, 9, 57, 37, 781, 13, 1093, 17, 3, 1, 1, 9, 10, 73, 50, 1555, 21, 5461, 82, 73, 1, 1, 1, 10, 11, 91, 65, 2801, 31, 19531, 257, 757, 11, 11, 1, 1, 11, 12, 111, 82, 4681, 43, 55987, 626, 4161, 61, 2047, 1, 1

Offset: 0

Eric Chen, Apr 22 2015

Outside of rows 0, 1, 2 and columns 0, 1, only terms of A206942 occur.
Conjecture: There are infinitely many primes in every row (except row 0) and every column (except column 0), the indices of the first prime in n-th row and n-th column are listed in A117544 and A117545. (See A206864 for all the primes apart from row 0, 1, 2 and column 0, 1.)
Another conjecture: Except row 0, 1, 2 and column 0, 1, the only perfect powers in this table are 121 (=Phi_5(3)) and 343 (=Phi_3(18)=Phi_6(19)).

			Read by antidiagonals:
m\n  0   1   2   3   4   5   6   7   8   9  10  11  12
------------------------------------------------------
0    1   1   1   1   1   1   1   1   1   1   1   1   1
1   -1   0   1   2   3   4   5   6   7   8   9  10  11
2    1   2   3   4   5   6   7   8   9  10  11  12  13
3    1   3   7  13  21  31  43  57  73  91 111 133 157
4    1   2   5  10  17  26  37  50  65  82 101 122 145
5    1   5  31 121 341 781 ... ... ... ... ... ... ...
6    1   1   3   7  13  21  31  43  57  73  91 111 133
etc.
The cyclotomic polynomials are:
n        n-th cyclotomic polynomial
0        1
1        x-1
2        x+1
3        x^2+x+1
4        x^2+1
5        x^4+x^3+x^2+x+1
6        x^2-x+1
...

Eric Chen, Table of n, a(n) for n = 0..5049 (first 100 antidiagonals)
Eric Weisstein's World of Mathematics, Cyclotomic polynomial
Wikipedia, Cyclotomic polynomial
Index entries for Cyclotomic polynomials, values at X

Rows 0-16 are A000012, A023443, A000027, A002061, A002522, A053699, A002061, A053716, A002523, A060883, A060884, A060885, A060886, A060887, A060888, A060889, A060890.
Columns 0-13 are A158388, A020500, A019320, A019321, A019322, A019323, A019324, A019325, A019326, A019327, A019328, A019329, A019330, A019331.
Main diagonal is A070518.
Indices of primes in n-th row for n = 1-20 are A008864, A006093, A002384, A005574, A049409, A055494, A100330, A000068, A153439, A246392, A162862, A246397, A217070, A250174, A250175, A006314, A217071, A164989, A217072, A250176.
Indices of primes in n-th column for n = 1-10 are A246655, A072226, A138933, A138934, A138935, A138936, A138937, A138938, A138939, A138940.
Indices of primes in main diagonal is A070519.
Cf. A117544 (indices of first prime in n-th row), A085398 (indices of first prime in n-th row apart from column 1), A117545 (indices of first prime in n-th column).
Cf. A206942 (all terms (sorted) for rows>2 and columns>1).
Cf. A206864 (all primes (sorted) for rows>2 and columns>1).

Table[Cyclotomic[m, k-m], {k, 0, 49}, {m, 0, k}]

t1(n)=n-binomial(floor(1/2+sqrt(2+2*n)), 2)
t2(n)=binomial(floor(3/2+sqrt(2+2*n)), 2)-(n+1)
T(m, n) = if(m==0, 1, polcyclo(m, n))
a(n) = T(t1(n), t2(n))

T(m, n) = Phi_m(n)

A000059 Numbers k such that (2k)^4 + 1 is prime.

Original entry on oeis.org

1, 2, 3, 8, 10, 12, 14, 17, 23, 24, 27, 28, 37, 40, 41, 44, 45, 53, 59, 66, 70, 71, 77, 80, 82, 87, 90, 97, 99, 102, 105, 110, 114, 119, 121, 124, 127, 133, 136, 138, 139, 144, 148, 156, 160, 164, 167, 170, 176, 182, 187, 207, 215, 218, 221, 233, 236, 238, 244, 246

Offset: 1

N. J. A. Sloane

			(2 * 2)^4 + 1 = 4^4 + 1 = 17, which is prime, so 2 is in the sequence.
(2 * 3)^4 + 1 = 6^4 + 1 = 1297, which is prime, so 3 is in the sequence.
(2 * 4)^4 + 1 = 8^4 + 1 = 4097 = 17 * 241, so 4 is not in the sequence.

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
J. Bohman, New primes of the form n^4+1, Nordisk Tidskr. Informationsbehandling (BIT) 13 (1973), 370-372.
M. Lal, Primes of the form n^4 + 1, Math. Comp., 21 (1967), 245-247.

Cf. A037896 (primes of the form n^4 + 1).

[n: n in [1..10000] | IsPrime((2*n)^4+1)] # Vincenzo Librandi, Nov 18 2010

A000059:=n->`if`(isprime((2*n)^4+1),n,NULL): seq(A000059(n), n=1..250); # Wesley Ivan Hurt, Aug 26 2014

Select[Range[300], PrimeQ[(2 * #)^4 + 1] &] (* Vladimir Joseph Stephan Orlovsky, Jan 24 2012 *)

for(n=1,10^3,if(isprime( (2*n)^4+1 ),print1(n,", "))) \\ Hauke Worpel (thebigh(AT)outgun.com), Jun 11 2008 [edited by Michel Marcus, Aug 27 2014]

from sympy import isprime
print([n for n in range(10**3) if isprime(16*n**4+1)])
# Derek Orr, Aug 27 2014

a(n) = A000068(n+1)/2 for n >= 1. [Corrected by Jianing Song, Feb 03 2019]

More terms from Hugo Pfoertner, Aug 27 2003

A067662 Numbers n such that n^2 + 1 and n^4 + 1 are primes.

Original entry on oeis.org

1, 2, 4, 6, 16, 20, 24, 54, 56, 74, 90, 160, 180, 204, 210, 340, 430, 436, 466, 556, 584, 690, 760, 930, 936, 966, 986, 1144, 1150, 1246, 1406, 1434, 1586, 1644, 1700, 1824, 1850, 1870, 1910, 2074, 2126, 2224, 2260, 2266, 2336, 2360, 2536, 2604, 2646, 2676

Offset: 1

Benoit Cloitre, Feb 23 2002

nonn

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 5000 terms from Seiichi Manyama)

Intersection of A000068 and A005574.

[n: n in [0..10000]| IsPrime(n^2+1) and IsPrime(n^4+1)] // Vincenzo Librandi, Aug 07 2010

A217796 Primes of the form n^4+1 such that (n+2)^4+1 is also prime.

Original entry on oeis.org

17, 257, 4477457, 8503057, 40960001, 59969537, 384160001, 5802782977, 58594980097, 94197431057, 102627966737, 114733948177, 283982410001, 330123790097, 381671897617, 405519334417, 691798081537, 741637881857, 1700843738897, 1749006250001, 2073600000001

Offset: 1

Michel Lagneau, Oct 12 2012

nonn

The corresponding n are in A217795.

			257 is in the sequence because  4^4+1 = 257 and (4+2)^4+1 = 1297 are both prime.

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

Cf. A000068, A002523, A037896, A217795.

for n from 0 by 2 to 3500 do: if type(n^4+1,prime)=true and type((n+2)^4+1,prime)=true then printf(`%d, `, n^4+1):else fi:od:

lst={}; Do[p=n^4+1; q=(n+2)^4+1;If[PrimeQ[p] && PrimeQ[q], AppendTo[lst, p]], {n, 0, 3500}];lst
Select[Partition[Table[n^4+1,{n,1300}],3,1],AllTrue[{#[[1]],#[[3]]}, PrimeQ]&][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 17 2020 *)

A235982 Numbers n of the form p^4 + 1 (for prime p) such that n^4 + 1 is also prime.

Original entry on oeis.org

82, 38950082, 47458322, 131079602, 1982119442, 25856961602, 58120048562, 602425897922, 1053022816562, 1267247769842, 3491998578722, 7181161893362, 7759350084722, 10756569837842, 16948379819282, 28424689653362, 33122338550402, 36562351115762, 50897394646082

Offset: 1

Derek Orr, Jan 17 2014

nonn

All numbers are congruent to 2 mod 20.

			10756569837842 = 1811^4 + 1 (1811 is prime) and 10756569837842^4 + 1 is prime, so 10756569837842 is a member of this sequence.

Cf. A002523, A000068.

nfp4Q[n_]:=Module[{p=Surd[n-1,4]},AllTrue[{p,n^4+1},PrimeQ]]; Select[ Range[ 2700]^4+ 1,nfp4Q] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 08 2019 *)

import sympy
from sympy import isprime
{print(n**4+1) for n in range(10000) if isprime(n) if isprime((n**4+1)**4+1)}

A337608 Decimal expansion of Lal's constant: the Hardy-Littlewood constant for A217795.

Original entry on oeis.org

7, 9, 2, 2, 0, 8, 2, 3, 8, 1, 6, 7, 5, 4, 1, 6, 6, 8, 7, 7, 5, 4, 5, 5, 5, 6, 6, 5, 7, 9, 0, 2, 4, 1, 0, 1, 1, 2, 8, 9, 3, 2, 2, 5, 0, 9, 8, 6, 2, 2, 1, 1, 1, 7, 2, 2, 7, 9, 7, 3, 4, 5, 2, 5, 6, 9, 5, 1, 4, 1, 5, 4, 9, 4, 4, 1, 2, 4, 9, 0, 6, 6, 0, 2, 9, 5, 3, 8, 8, 3, 9, 8, 0, 2, 7, 5, 2, 9, 2, 7, 8, 7, 3, 9, 7, 3

Offset: 0

Amiram Eldar, Sep 04 2020

Shanks (1967) conjectured that the number of primes of the form (m + 1)^4 + 1 such that (m - 1)^4 + 1 is also a prime (A217795 plus 1), with m <= x, is asymptotic to c * li_2(x), where li_2(x) = Integral_{t=2..n} (1/log(t)^2) dt, and c is this constant. He defined c as in the formula section, evaluated it by 0.79220 and named it after the mathematician Mohan Lal, who conjectured the asymptotic formula without evaluating this constant.

The first 100 digits of this constant were calculated by Ettahri et al. (2019).

			0.792208238167541668775455566579024101128932250986221...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, pp. 90-91.

Keith Conrad, Hardy-Littlewood constants in: Mathematical properties of sequences and other combinatorial structures, Jong-Seon No et al. (eds.), Kluwer, Boston/Dordrecht/London, 2003, pp. 133-154, alternative link.
Salma Ettahri, Olivier Ramaré, Léon Surel, Fast multi-precision computation of some Euler products, arXiv:1908.06808 [math.NT], 2019 (Corollary 1.9).
Mohan Lal, Primes of the form n^4 + 1, Mathematics of Computation, Vol. 21, No. 98 (1967), pp. 245-247.
Daniel Shanks, Lal's constant and generalizations, Mathematics of Computation, Vol. 21, No. 100 (1967), pp. 705-707.
Eric Weisstein's World of Mathematics, Lal's Constant.

Cf. A000068, A037896, A088367, A210630, A217795, A217796.

Similar constants: A005597, A331941, A337606, A337607.

$MaxExtraPrecision = 1000; digits = 121;
f[p_] := (p-8)*(p+1)^4/((p-1)^4*p);
coefs = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, 1000}], x]];
S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums);
P[m_, n_, s_] := 1/EulerPhi[m] * Sum[Conjugate[DirichletCharacter[m, r, n]] * S[m, r, s], {r, 1, EulerPhi[m]}] + Sum[If[GCD[p, m] > 1 && Mod[p, m] == n, 1/p^s, 0], {p, 1, m}];
m = 2; sump = 0; difp = 1; While[Abs[difp] > 10^(-digits - 5) || difp == 0, difp = coefs[[m]]*(P[8, 1, m] - 1/17^m); sump = sump + difp; m++];
RealDigits[Chop[N[f[17] * Pi^4/(2^7 * Log[1+Sqrt[2]]^2) * Exp[sump], digits]], 10, digits - 1][[1]] (* Vaclav Kotesovec, Jan 16 2021 *)

Equals (Pi^4/(2^7 * log(1+sqrt(2))^2)) * Product_{primes p == 1 (mod 8)} (1 - 4/p)^2 * ((p + 1)/(p - 1))^4 * p*(p-8)/(p-4)^2 = (Pi^2/32) * A088367^2 * A334826^2 * A210630 = 2 * A337607^2 * A210630.

More terms from Vaclav Kotesovec, Jan 16 2021

A073476 Numbers k such that k^4 + 1, (k+2)^4 + 1 and (k+4)^4 + 1 are all primes.

Original entry on oeis.org

2, 2222, 2732, 3998, 5356, 5358, 5626, 8034, 9402, 9972, 10006, 10930, 12188, 12322, 12702, 13372, 14536, 15038, 15962, 21396, 24704, 25446, 27118, 29566, 36126, 36604, 36732, 36734, 37550, 37552, 37554, 44176, 44218, 48164, 48978

Offset: 1

Martin Raab, Aug 26 2002

nonn

			2222^4+1, 2224^4+1 and 2226^4+1 are prime

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A000068, n such that n^4+1 is prime.

N:= 10^5: # to get all terms <= N
R:= select(t -> isprime(t^4+1), [seq(i,i=2..N,2)]):
V:= select(i -> R[i+2]=R[i]+4, [$1..nops(R)-2]):
R[V]; # Robert Israel, Apr 20 2017

Select[Range[5000], PrimeQ[ #^4 + 1] && PrimeQ[(# + 2)^4 + 1] && PrimeQ[(# + 4)^4 + 1] & ]

More terms from Robert G. Wilson v, Aug 28 2002

A087738 Square array: T(n,k) gives n-th number a such that a^(2^k)+1 is prime (a generalized Fermat).

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 6, 4, 2, 1, 10, 6, 4, 2, 1, 12, 10, 6, 4, 2, 1, 16, 14, 16, 118, 44, 30, 1, 18, 16, 20, 132, 74, 54, 102, 1, 22, 20, 24, 140, 76, 96, 162, 120, 1, 28, 24, 28, 152, 94, 112, 274, 190, 278, 1, 30, 26, 34, 208, 156, 114, 300, 234, 614, 46, 1, 36, 36, 46, 240, 158

Offset: 0

Jeppe Stig Nielsen, Oct 01 2003

			{1}; {2,1}; {4,2,1}; ...
See the well-formed array on Gallot's page.

Harvey Dubner, J. Recr. Math., 18, 1986.

Yves Gallot, Generalized Fermat Prime Search.

Cf. A056993, A006093, A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362.

A121979 Numbers k such that (2*k^2)^4 + 1 is prime.

Original entry on oeis.org

1, 11, 12, 20, 24, 27, 28, 34, 40, 44, 61, 74, 79, 82, 95, 96, 119, 131, 136, 147, 148, 156, 164, 170, 173, 180, 187, 209, 211, 238, 252, 255, 269, 279, 299, 328, 337, 340, 343, 371, 379, 380, 388, 397, 413, 452, 462, 473, 476, 483, 516, 522, 527, 530, 539, 572

Offset: 1

Alexander Adamchuk, Sep 10 2006

Corresponding primes of the form (2*k^2)^4 + 1 are {17, 3429742097, 6879707137, 409600000001, ...}.

There are consecutive twin pairs {a(n),a(n+1)} = {11,12}, {27,28}, {95,96},{147,148}, ...

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000068.

Select[Range[1000],PrimeQ[(2*#1^2)^4+1]&]

is(n)=isprime((2*n^2)^4+1) \\ Charles R Greathouse IV, Jun 13 2017

A217129 Numbers n such that n^4 + 1 is not prime.

Original entry on oeis.org

3, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 81, 83

Offset: 1

Vincenzo Librandi, Sep 27 2012

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

Cf. A000068, A037896.

Magma
```
[n: n in [1..100] |not IsPrime(n^4+1)];
```
Mathematica
```
Select[Range[100], ! PrimeQ[#^4 + 1] &]
```

cp's OEIS Frontend

A253240 Square array read by antidiagonals: T(m, n) = Phi_m(n), the m-th cyclotomic polynomial at x=n.

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A000059 Numbers k such that (2k)^4 + 1 is prime.

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A067662 Numbers n such that n^2 + 1 and n^4 + 1 are primes.

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A217796 Primes of the form n^4+1 such that (n+2)^4+1 is also prime.

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A235982 Numbers n of the form p^4 + 1 (for prime p) such that n^4 + 1 is also prime.

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A337608 Decimal expansion of Lal's constant: the Hardy-Littlewood constant for A217795.

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A073476 Numbers k such that k^4 + 1, (k+2)^4 + 1 and (k+4)^4 + 1 are all primes.

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