A037896 -id:A037896 - cp's OEIS frontend

A193562 Number of divisors of n^4+1.

1, 2, 2, 4, 2, 4, 2, 4, 4, 8, 4, 4, 4, 4, 4, 8, 2, 4, 4, 8, 2, 4, 4, 4, 2, 8, 4, 8, 2, 4, 4, 8, 4, 8, 2, 4, 4, 8, 4, 4, 4, 8, 4, 16, 8, 8, 2, 8, 2, 8, 4, 8, 4, 8, 2, 8, 2, 4, 4, 16, 8, 4, 4, 8, 8, 4, 8, 8, 4, 8, 8, 4, 4, 4, 2, 8, 8, 16, 4, 16, 2, 4, 2, 16, 4

Offset: 0

Jonathan Vos Post, Aug 09 2011

This is to n^4+1 as A193432 is to n^2+1.

a(n) = 2 when n^4+1 is prime, iff n is in A037896.

			a(3) = 4 because 3^4+1 = 82, whose 4 factors are {1, 2, 41, 82}.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000005, A002523, A037896, A193432 (number of divisors of n^2+1).

[NumberOfDivisors(n^4+1):n in [0..90]]; // Marius A. Burtea, Feb 09 2020

DivisorSigma[0,Range[0,90]^4+1] (* Harvey P. Dale, May 05 2013 *)

a(n) = numdiv(n^4+1); \\ Michel Marcus, Feb 09 2020

a(n) = A000005(A002523(n)) = d(n^4+1) (also called tau(n^4+1) or sigma_0(n^4+1)), the number of divisors of n^4+1.

A193929 Number of prime factors of n^4 + 1, counted with multiplicity.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 2, 3, 2, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 2, 2, 2, 1, 3, 2, 3, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 2, 2, 2, 3, 2, 4, 3, 3, 1, 3, 1, 3, 2, 3, 2, 3, 1, 3, 1, 2, 2, 4, 3, 2, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 2, 2, 1, 3, 3, 4, 2, 4, 1, 2, 1, 4, 2, 4, 2, 3, 1, 3, 1, 3, 2, 3, 2, 4, 3, 3, 2, 3, 2, 3, 2, 2, 3, 2, 1, 3, 2, 3, 3, 4, 2, 2, 2, 2, 2, 3, 1

Offset: 0

Jonathan Vos Post, Aug 09 2011

This is to A193330 as A002523(n) = n^4+1 is to A002522(n) = n^2 + 1. a(n) = 2 when n^4+1 is prime, iff n is in A037896.

			a(9) = 3 because 9^4+1 = 6562 = 2 * 17 * 193, which has 3 prime factors, counted with multiplicity

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A001222, A002523, A193432, A193562.

[0] cat [&+[p[2]: p in Factorization(n^4+1)]:n in [1..120]]; // Marius A. Burtea, Feb 09 2020

Join[{0}, Table[Total[Transpose[FactorInteger[n^4 + 1]][[2]]], {n, 100}]] (* T. D. Noe, Aug 10 2011 *)
Join[{0},Table[PrimeOmega[n^4+1],{n,120}]] (* Harvey P. Dale, Sep 25 2012 *)

a(n) = bigomega(n^4+1); \\ Michel Marcus, Feb 09 2020

a(n) = A001222(A002523(n)) = bigomega(n^4+1) or Omega(n^4+1).

A217795 Numbers n such that n^4+1 and (n+2)^4+1 are both prime.

Original entry on oeis.org

2, 4, 46, 54, 80, 88, 140, 276, 492, 554, 566, 582, 730, 758, 786, 798, 912, 928, 1142, 1150, 1200, 1236, 1404, 1540, 1552, 1610, 1644, 1650, 1932, 1942, 2044, 2102, 2204, 2222, 2224, 2238, 2254, 2374, 2436, 2486, 2510, 2640, 2674, 2698, 2732, 2734, 3244, 3286

Offset: 1

Michel Lagneau, Oct 12 2012

			4 is in the sequence because 4^4+1 = 257 and 6^4+1 = 1297 are both prime.

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

Cf. A000068, A002523, A037896.

[n: n in [0..3300] | IsPrime(n^4 + 1) and IsPrime((n + 2)^4 + 1)]; // Vincenzo Librandi, Oct 13 2012

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):else fi:od:

lst={}; Do[p=n^4+1; q=(n+2)^4+1;If[PrimeQ[p] && PrimeQ[q], AppendTo[lst, n]], {n, 0, 3000}];lst
Select[Range[3500],AllTrue[{#^4+1,(#+2)^4+1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 29 2015 *)

A230261 Number of ways to write 2*n - 1 = p + q with p, p + 6 and q^4 + 1 all prime, where q is a positive integer.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Oct 14 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 3. Also, any odd number greater than 6 can be written as p + q (q > 0) with p, p + 6 and q^2 + 1 all prime.
(ii) Any integer n > 1 can be written as x + y (x, y > 0) with x^4 + 1 and y^2 + y + 1 both prime.
(iii) Each integer n > 2 can be expressed as x + y (x, y > 0) with 4*x^2 + 3 and 4*y^2 -3 both prime.
Either of parts (i) and (ii) implies that there are infinitely many primes of the form x^4 + 1.

			a(6) = 2 since 2*6-1 = 5 + 6 = 7 + 4, and 5, 5+6 = 11, 7, 7+6 = 13, 6^4+1 = 1297 and 4^4+1 = 257 are all prime.
a(25) = 1 since 2*25-1 = 47 + 2, and 47, 47+6 = 53, 2^4+1 = 17 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Cf. A000068, A023201, A037896, A219864, A227908, A227909, A230241, A230252, A230254.

a[n_]:=Sum[If[PrimeQ[Prime[i]+6]&&PrimeQ[(2n-1-Prime[i])^4+1],1,0],{i,1,PrimePi[2n-2]}]
Table[a[n],{n,1,100}]

a(n)=my(s,p=5,q=7);forprime(r=11,2*n+4,if(r-p==6&&isprime((2*n-1-p)^4+1),s++); if(r-q==6&&isprime((2*n-1-q)^4+1),s++); p=q;q=r);s \\ Charles R Greathouse IV, Oct 14 2013

A258805 Primes of the form k^8 + 1.

Original entry on oeis.org

2, 257, 65537, 37588592026706177, 92170395205042177, 147578905600000001, 284936905588473857, 3503536769037500417, 11007531417600000001, 11763130845074473217, 47330370277129322497, 50024641296100000001, 76872571987558646017, 416806419029812551937

Offset: 1

Vincenzo Librandi, Jun 11 2015

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

Subsequence of A002496, A037896.

Cf. A006314 (associated n), A060890.

[a: n in [1..500] | IsPrime(a) where a is n^8+1];

Mathematica
```
Select[Range[500]^8 + 1, PrimeQ]
```

is(n)=ispower(n-1,8) && isprime(n) \\ Charles R Greathouse IV, Jun 11 2015

a(n) = A060890(A006314(n)). - Michel Marcus, Jun 11 2015

A307690 Integers with only one prime factor and whose Euler's totient is a perfect biquadrate.

Original entry on oeis.org

2, 17, 32, 257, 512, 1297, 8192, 65537, 131072, 160001, 331777, 614657, 1336337, 1419857, 2097152, 4477457, 5308417, 8503057, 9834497, 29986577, 33554432, 40960001, 45212177, 59969537, 65610001, 126247697, 193877777, 303595777, 384160001, 406586897, 536870912, 562448657, 655360001

Offset: 1

Bernard Schott, Apr 22 2019

nonn

An integer q is a term iff q = p^(4*m+1), when p is prime of the form k^4 + 1 and m >= 0, then phi(q) = (k * (k^4+1)^m)^4. The primitive terms of this sequence are the primes of the form p = k^4 + 1, which are exactly in A037896.

			a(14) = 1419857 = 17^5 and phi(1419857) = 34^4.

Subsequences: A013776 (2^(4*m+1)), A013806 (17^(4*m+1)), A037896 (primes of the form k^4 + 1).
Intersection of A078164 and A246655.
Cf. A054755 (idem with Euler's totient is square).

[n:n in [1..10000000]| #PrimeDivisors(n) eq 1 and IsPower(EulerPhi(n),4)]; // Marius A. Burtea, May 09 2019

isok(n) = isprimepower(n) && ispower(eulerphi(n), 4); \\ Michel Marcus, Apr 23 2019

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

A182344 Primes of the form n^4 + 3.

Original entry on oeis.org

3, 19, 4099, 65539, 234259, 456979, 614659, 1336339, 3748099, 14776339, 21381379, 33362179, 45212179, 71639299, 116985859, 146410003, 193877779, 268435459, 322417939, 759333139, 1146228739, 1664966419, 2019963139

Offset: 1

Patrick Devlin, Apr 25 2012

nonn

			3 = 0^4 + 3; 19 = 2^4 + 3.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A037896.

# choose N large, then S is the desired set
f:=n->n^4 + 3:
S:={}:
for n from 0 to N do if(isprime(f(n))) then S:=S union {f(n)}: fi: od:

Select[Range[0,250]^4+3,PrimeQ] (* Harvey P. Dale, Jan 10 2013 *)

A182345 Primes of the form n^4 + 5.

Original entry on oeis.org

5, 1301, 331781, 18974741, 37015061, 136048901, 429981701, 1196883221, 1731891461, 4032758021, 4430766101, 12745506821, 13680577301, 43237380101, 74247530261, 92844527621, 151613669381, 196741925141

Offset: 1

Patrick Devlin, Apr 25 2012

nonn

All terms == 5 (mod 24). All terms except the first == 101 mod 120. - Robert Israel, Jul 01 2019

			5 = 1^4 + 5; 1301 = 6^4 + 5.

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

Cf. A037896.

# choose N large, then S is the desired set
f:=n->n^4 + 5:
S:={}:
for n from 0 to N do if(isprime(f(n))) then S:=S union {f(n)}: fi: od:

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 *)

cp's OEIS Frontend

A193562 Number of divisors of n^4+1.

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A193929 Number of prime factors of n^4 + 1, counted with multiplicity.

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A217795 Numbers n such that n^4+1 and (n+2)^4+1 are both prime.

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A230261 Number of ways to write 2*n - 1 = p + q with p, p + 6 and q^4 + 1 all prime, where q is a positive integer.

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A258805 Primes of the form k^8 + 1.

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A307690 Integers with only one prime factor and whose Euler's totient is a perfect biquadrate.

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

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