A037896 -id:A037896 - cp's OEIS frontend

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

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).

A078164 Numbers k such that phi(k) is a perfect biquadrate.

Original entry on oeis.org

1, 2, 17, 32, 34, 40, 48, 60, 257, 512, 514, 544, 640, 680, 768, 816, 960, 1020, 1297, 1387, 1417, 1729, 1971, 2109, 2223, 2289, 2331, 2445, 2457, 2565, 2594, 2608, 2774, 2812, 2834, 2835, 3052, 3260, 3458, 3888, 3912, 3924, 3942, 3996, 4104, 4212, 4218

Offset: 1

Labos Elemer, Nov 27 2002

nonn

Corresponding values of phi include 1, 16, 256, 1296, 4096, ... and these arise several times each.
a(3) = A053576(4).
A013776 is a subsequence since phi(2^(4*n+1)) = (2^n)^4. - Bernard Schott, Sep 22 2022
Subsequence of primes is A037896 since in this case: phi(k^4+1) = k^4. - Bernard Schott, Mar 05 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A039770. A037896 is a subsequence.
Sequences where phi(k) is a perfect power: A039770 (square), A039771 (cube), this sequence (4th), A078165 (5th), A078166 (6th), A078167 (7th), A078168 (8th), A078169 (9th), A078170 (10th).
Cf. A001317, A053576, A037896, A045544, A000010, A013776.

k=4; Do[s=EulerPhi[n]^(1/k); If[IntegerQ[s], Print[n]], {n, 1, 5000}]
Select[Range[5000],IntegerQ[Surd[EulerPhi[#],4]]&] (* Harvey P. Dale, Apr 30 2015 *)

is(n)=ispower(eulerphi(n),4) \\ Charles R Greathouse IV, Apr 24 2020

from itertools import count, islice
from sympy import totient, integer_nthroot
def A078164_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:integer_nthroot(totient(n),4)[1], count(max(1,startvalue)))
A078164_list = list(islice(A078164_gen(),20)) # Chai Wah Wu, Feb 28 2023

A096172 Largest prime factor of n^4 + 1.

Original entry on oeis.org

2, 17, 41, 257, 313, 1297, 1201, 241, 193, 137, 7321, 233, 14281, 937, 1489, 65537, 41761, 929, 3833, 160001, 97241, 3209, 139921, 331777, 11489, 26881, 6481, 614657, 353641, 3361, 1129, 61681, 6113, 1336337, 750313, 98801, 10529, 50857, 1156721

Offset: 1

Hugo Pfoertner, Jun 19 2004

nonn

Mabkhout shows that a(n) >= 137 for n > 3. - Charles R Greathouse IV, Apr 07 2014

			a(1)=2 because 1^4 + 1 = 2;
a(2)=17: 2^4 + 1 = 17;
a(8)=241: 8^4 + 1 = 4097 = 17*241.

Mustapha Mabkhout, Minoration de P(x^4+1), Rendiconti del Seminario della Facoltà di Scienze dell'Università di Cagliari 63:2 (1993), pp. 135-148.

Vincenzo Librandi and T. D. Noe, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)
Florian Luca, Primitive divisors of Lucas sequences and prime factors of x^2 + 1 and x^4 + 1, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004), pp. 19-24.
Marina Mureddu, A lower bound for P(x^4+1), Annales de la Faculté des sciences de Toulouse: Mathématiques, Serie 5, Vol. 8, No. 2 (1986-1987), pp. 109-119.

Cf. A000068, A002523, A014442, A037896, A006530, A069170, A096169, A096171.

FactorInteger[#^4+1][[-1,1]]&/@Range[40] (* Harvey P. Dale, Apr 30 2012 *)

a(n)=my(f=factor(n^4+1)[,1]); f[#f] \\ Charles R Greathouse IV, Apr 07 2014

a(n) = A006530(1+n^4) = A014442(n^2). - R. J. Mathar, Jan 28 2017
From Amiram Eldar, Oct 28 2024: (Start)
a(n) > 113 for n > 3 (Mureddu, 1986-1987).
a(n) >= 233 for n >= 11 (Luca, 2004). (End)

A057447 a(n+1) = next prime such that a(n+1)-1 | (a(1)...a(n))^3.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 101, 107, 109, 127, 131, 139, 149, 151, 157, 167, 173, 179, 181, 191, 197, 199, 211, 223, 229, 233, 251, 263, 269, 271, 277, 281, 283, 293, 311, 313, 317, 331, 347, 349, 359

Offset: 1

Robert G. Wilson v, Sep 25 2000

nonn

No prime of the form a*b^k + 1, with a > 0, b > 1 and k > 3 (including those in A037896) belongs to the sequence. - Mauro Fiorentini, Aug 09 2023

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

Cf. A007459, A037896, A057448.

NextPrime[ n_Integer ] := Module[ {k = n + 1}, While[ ! PrimeQ[ k ], k++ ]; Return[ k ] ]; f[ n_List ] := (a = n; b = Apply[ Times, a^3 ]; d = NextPrime[ a[[ -1 ]] ]; While[ ! IntegerQ[ b/(d - 1) ] && d < b+2, d = NextPrime[ d ] ]; AppendTo[ a, d ]; Return[ a ]); Nest[ f, {2}, 75 ]

A096169 Odd n such that (n^4+1)/2 is prime.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 21, 23, 29, 35, 39, 57, 61, 65, 71, 73, 81, 103, 105, 113, 115, 119, 129, 153, 165, 169, 171, 199, 203, 205, 251, 259, 267, 275, 309, 313, 317, 333, 337, 339, 353, 363, 403, 405, 415, 419, 431, 445, 449, 453, 455, 463, 471, 477, 479, 487

Offset: 1

Hugo Pfoertner, Jun 19 2004

nonn

			a(1)=3 because (3^4+1)/2=82/2=41 is prime.

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

Cf. A000068 n^4+1 is prime, A037896 primes of the form n^4+1, A096170 primes of the form (n^4+1)/2, A096171 n^4+1 is an odd semiprime, A096172 largest prime factor of n^4+1.

[ n: n in [0..2500] | IsPrime((n^4+1) div 2) ]; // Vincenzo Librandi, Apr 15 2011

Select[Range[1,501,2],PrimeQ[(#^4+1)/2]&] (* Harvey P. Dale, Jun 04 2011 *)

A096171 Numbers k such that k^4+1 is an odd semiprime.

Original entry on oeis.org

8, 10, 12, 14, 18, 22, 26, 30, 32, 36, 38, 40, 42, 50, 52, 58, 62, 68, 72, 78, 84, 86, 92, 94, 98, 100, 102, 108, 112, 114, 116, 120, 122, 124, 128, 130, 138, 146, 148, 152, 158, 162, 166, 170, 172, 176, 184, 186, 200, 212, 214, 216, 218, 222, 224, 226, 234, 250, 252

Offset: 1

Hugo Pfoertner, Jun 19 2004

nonn

			a(1)=8 because 8^4 + 1 = 4097 = 17*241;
a(2)=10: 10^4 + 1 = 10001 = 73*137.

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

Cf. A000068 (n^4+1 is prime), A037896 (primes of the form k^4+1), A096169 ((n^4+1)/2 is prime), A069170 (primes of the form (k^4+1)/2), A096172 (largest prime factor of n^4+1), A046388.

Select[Range[2,300,2],PrimeOmega[#^4+1]==2&] (* Harvey P. Dale, Dec 25 2021 *)

isA096171(n) = {local(m);m=n^4+1;(m%2==1)&&(bigomega(m)==2)} \\ Michael B. Porter, Feb 02 2010

A233549 Number of ways to write n = p + q (q > 0) with p and (phi(p)*phi(q))^4 + 1 prime, where phi(.) is Euler's totient function (A000010).

Original entry on oeis.org

0, 0, 1, 2, 2, 3, 3, 2, 3, 2, 1, 3, 1, 4, 3, 3, 4, 4, 6, 1, 1, 1, 4, 1, 2, 2, 4, 4, 1, 6, 7, 3, 4, 3, 4, 3, 3, 5, 2, 3, 5, 3, 1, 3, 5, 3, 3, 5, 6, 4, 4, 5, 4, 3, 4, 6, 4, 4, 3, 4, 5, 4, 2, 2, 4, 3, 6, 1, 4, 2, 8, 9, 2, 5, 5, 4, 2, 3, 4, 3, 6, 1, 7, 5, 8, 5, 4, 4, 4, 10, 10, 6, 4, 8, 4, 3, 4, 6, 6, 2

Offset: 1

Zhi-Wei Sun, Dec 12 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 2.
(ii) If n > 2 is not equal to 26, then there is a prime p < n with (phi(p)*phi(n-p))^2 + 1 prime.
(iii) If n > 3 is different from 9 and 16, then there is a prime p < n with ((p+1)*phi(n-p))^2 + 1 prime.
Part (i) of the conjecture implies that there are infinitely many primes of the form x^4 + 1. We have verified it for n up to 10^7.

			a(11) = 1 since 11 = 2 + 9 with 2 and (phi(2)*phi(9))^4 + 1 = 6^4 + 1 = 1297 both prime.
a(13) = 1 since 13 = 5 + 8 with 5 and (phi(5)*phi(8))^4 + 1 = 16^4 + 1 = 65537 both prime.
a(258) = 1 since 258 = 167 + 91 with 167 and (phi(167)*phi(91))^4 + 1 = (166*72)^4 + 1 = 20406209352892417 both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A000068, A037896, A233542, A233544, A233547.

a[n_]:=Sum[If[PrimeQ[((Prime[k]-1)*EulerPhi[n-Prime[k]])^4+1],1,0],{k,1,PrimePi[n-1]}]
Table[a[n],{n,1,100}]

A096170 Primes of the form (k^4 + 1)/2.

Original entry on oeis.org

41, 313, 1201, 7321, 14281, 41761, 97241, 139921, 353641, 750313, 1156721, 5278001, 6922921, 8925313, 12705841, 14199121, 21523361, 56275441, 60775313, 81523681, 87450313, 100266961, 138461441, 273990641, 370600313, 407865361

Offset: 1

Hugo Pfoertner, Jun 19 2004

nonn

Note that k must be odd. Terms of primitive Pythagorean triples: (k^2, (k^4-1)/2, (k^4+1)/2).

			a(1)=41 because (3^4 + 1)/2 = 82/2 = 41 is prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A096169 (n^4+1)/2 is prime, A000068 n^4+1 is prime, A037896 primes of the form n^4+1, A096171 n^4+1 is an odd semiprime, A096172 largest prime factor of n^4+1.

[ a: n in [0..2500] | IsPrime(a) where a is ((n^4+1) div 2) ]; // Vincenzo Librandi, Apr 15 2011

Select[(Range[200]^4+1)/2,PrimeQ] (* Harvey P. Dale, Mar 09 2013 *)

list(lim)=my(v=List(),t); forstep(n=3,sqrtnint(lim\1*2-1,4),2, if(isprime(t=(n^4+1)/2), listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Feb 14 2017

Name edited by Zak Seidov, Apr 14 2011

A182343 Primes of the form k^4 + 2 for k >= 0.

Original entry on oeis.org

2, 3, 83, 6563, 50627, 194483, 4100627, 10556003, 15752963, 22667123, 57289763, 96059603, 276922883, 395254163, 6059221283, 6597500627, 12296370323, 14166950627, 42110733683, 44386483763, 46753250627, 49213429283, 69257922563, 72555348323

Offset: 1

Patrick Devlin, Apr 25 2012

Apart from 3, this is a subsequence of A053786. - Bruno Berselli, Apr 26 2012

			2 = 0^4 + 2;
3 = 1^4 + 2;
83 = 3^4 + 2.

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

Cf. A037896, A053786.

[2,3] cat [n^4+2: n in [3..600 by 6]|IsPrime(n^4+2)]; // Bruno Berselli, Apr 26 2012

# choose N large, then S is the desired set
f:=n->n^4 + 2:
S:={}:
for n from 0 to N do if(isprime(f(n))) then S:=S union {f(n)}: fi: od:

Select[Range[0,600]^4 + 2,PrimeQ] (* Bruno Berselli, Apr 26 2012 *)

A188717 Primes p such that all prime factors of p-1 have exponent 4.

Original entry on oeis.org

2, 17, 1297, 1336337, 4477457, 29986577, 45212177, 126247697, 193877777, 406586897, 562448657, 916636177, 1416468497, 1944810001, 3208542737, 4162314257, 5006411537, 5972816657, 12444741137, 19565295377, 34188010001, 38167092497, 47156728337, 59553569297, 61505984017

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 09 2011

nonn

			17-1 = 2^4, 1297-1 = 2^4*3^4, 1336337-1 = 2^4*17^4, 4477457-1 = 2^4*23^4, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A089195 (exponent 2), A037896 (primes of the form k^4+1), A188764.

Prepend[Select[Table[Prime[n],{n,600000}],Length[Union[Last/@FactorInteger[#-1]]]==1&&Union[Last/@FactorInteger[#-1]]=={4}&], 2]
seq[lim_] := Select[Select[Range[Floor[Surd[lim-1, 2]]], SquareFreeQ]^4 + 1, PrimeQ]; seq[10^6] (* Amiram Eldar, Jan 18 2025 *)

list(lim) = select(isprime, apply(x -> x^4 + 1, select(issquarefree, vector(sqrtnint(lim-1, 4), i, i)))); \\ Amiram Eldar, Jan 18 2025

a(12)-a(22) from Donovan Johnson, Apr 10 2011

a(1) = 2 inserted and a(23)-a(25) added by Amiram Eldar, Jan 18 2025

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

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A078164 Numbers k such that phi(k) is a perfect biquadrate.

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A096172 Largest prime factor of n^4 + 1.

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A057447 a(n+1) = next prime such that a(n+1)-1 | (a(1)...a(n))^3.

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A096169 Odd n such that (n^4+1)/2 is prime.

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A096171 Numbers k such that k^4+1 is an odd semiprime.

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A233549 Number of ways to write n = p + q (q > 0) with p and (phi(p)*phi(q))^4 + 1 prime, where phi(.) is Euler's totient function (A000010).

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A096170 Primes of the form (k^4 + 1)/2.

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