A002645 -id:A002645 - cp's OEIS frontend

A233567 Number of ways to write n = p + q (q > 0) with p and p^4 + phi(q)^4 both prime, where phi(.) is Euler's totient function (A000010).

0, 0, 1, 1, 0, 1, 1, 2, 2, 2, 3, 1, 3, 2, 4, 2, 3, 4, 3, 4, 5, 3, 5, 2, 6, 4, 3, 4, 5, 2, 1, 2, 3, 5, 5, 1, 3, 3, 4, 3, 3, 7, 6, 4, 7, 2, 5, 5, 5, 5, 3, 7, 4, 7, 4, 6, 5, 3, 5, 6, 6, 5, 5, 8, 9, 6, 7, 5, 6, 5, 7, 7, 5, 8, 7, 6, 6, 6, 8, 8, 5, 8, 11, 3, 7, 6, 7, 8, 7, 1, 8, 5, 6, 9, 10, 8, 9, 12, 8, 6

Offset: 1

Zhi-Wei Sun, Dec 13 2013

nonn

Conjecture: If n > 2 is not equal to 5, then we have a(n) > 0, also there is a prime p < n with p^2 + phi(n-p)^2 prime.

We have verified this for n up to 10^7. The first assertion in the conjecture implies that there are infinitely many primes of the form p^4 + q^4, where p is a prime and q is a positive integer.

			a(7) = 1 since 7 = 3 + 4 with 3 and 3^4 + phi(4)^4 = 81 + 16 = 97 both prime.
a(12) = 1 since 12 = 7 + 5 with 7 and 7^4 + phi(5)^4 = 7^4 + 4^4 = 2657 both prime.
a(31) = 1 since 31 = 23 + 8 with 23 and 23^4 + phi(8)^4 = 23^4 + 4^4 = 280097 both prime.
a(36) = 1 since 36 = 3 + 33 with 3 and 3^4 + phi(33)^4 = 3^4 + 20^4 = 160081 both prime.
a(90) = 1 since 90 = 79 + 11 with 79 and 79^4 + phi(11)^4 = 79^4 + 10^4 = 38960081 both prime.

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

Cf. A000010, A000040, A002645, A233542, A233544, A233547, A233549, A233566.

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

A100271 Primes of the form a^4 + b^3 with b>0.

Original entry on oeis.org

2, 17, 43, 89, 257, 283, 359, 593, 599, 1297, 2213, 2617, 3391, 3631, 4129, 4177, 4721, 6569, 7561, 8081, 8233, 9277, 10343, 10657, 10729, 11273, 12197, 13049, 13463, 14449, 14561, 15641, 15881, 16369, 16921, 17209, 17657, 19699, 22067, 24137

Offset: 1

T. D. Noe, Nov 18 2004

nonn

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

Cf. A002645 (primes of the form a^4 + b^4), A028916 (primes of the form a^4 + b^2), A100291 (numbers of the form a^4 + b^3).

lst={}; Do[p=a^4+b^3; If[p<50000&&PrimeQ[p], AppendTo[lst, p]], {a, 64}, {b, 256}]; Union[lst]

list(lim)=my(v=List([2]),a4,t); lim\=1; for(a=1,sqrtnint(lim-1,4), a4=a^4; forstep(b=1+a%2,sqrtnint(lim-a4,3),2, if(isprime(t=a4+b^3), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Aug 18 2017

A216285 Primes which cannot be written as x^4+y^4.

Original entry on oeis.org

3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397

Offset: 1

V. Raman, Sep 03 2012

nonn

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

Cf. A002645, A002144.

T=thueinit('x^4+1,1);
is(n)=#thue(T,n)==0&&isprime(n) \\ Charles R Greathouse IV, Dec 18 2014

a(n) ~ n log n. - Charles R Greathouse IV, Dec 18 2014

A247857 Primes of the form a^2 + b^4, with repetition.

Original entry on oeis.org

2, 5, 17, 17, 37, 41, 97, 97, 101, 137, 181, 197, 241, 257, 257, 277, 281, 337, 337, 401, 457, 577, 617, 641, 641, 661, 677, 757, 769, 821, 857, 881, 881, 977, 1097, 1109, 1201, 1217, 1237, 1297, 1297, 1301, 1321, 1409, 1481, 1601, 1657, 1697, 1777, 2017, 2069, 2137, 2281, 2389, 2417, 2417, 2437

Offset: 1

Jean-François Alcover, Sep 25 2014

nonn

Duplicates, which begin 17, 97, 257, 337, etc, are quartan primes A002645, except 2 (noticed by Michel Marcus).
Is there any triple?
No, by the uniqueness part of Fermat's two-squares theorem, at most one duplicate of a^2 + b^4 can exist. Namely, when a is a square, say a = B^2, then a^2 + b^4 = A^2 + B^4 where A = b^2. (This also proves Marcus's comment, since a^2 + b^4 = b^4 + B^4.) - Jonathan Sondow, Oct 03 2015

			Since 97 = 4^2 + 3^4 = 9^2 + 2^4, it appears twice in the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Art of Problem Solving, Fermat's Two Squares Theorem
John Friedlander and Henryk Iwaniec, Using a parity-sensitive sieve to count prime values of a polynomial, PNAS, vol. 94 no. 4, pp. 1054-1058.
Marek Wolf, Continued fractions constructed from prime numbers, arXiv:1003.4015 [math.NT], 2010, p. 8.
Wikipedia, Friedlander-Iwaniec theorem

Cf. A002645, A028916 (same sequence without repetition).

Cf. A000040, A256852.

a247857 n = a247857_list !! (n-1)
a247857_list = concat $ zipWith replicate a256852_list a000040_list
-- Reinhard Zumkeller, Apr 11 2015

max = 10^4; r = Reap[Do[n = a^2 + b^4; If[n <= max && PrimeQ[n], Sow[n]], {a, Sqrt[max]}, {b, max^(1/4)}]][[2, 1]]; Union[r, SameTest -> (False&)]

A182198 Primes of form a^2 + b^2 such that a^4 + b^4 is prime.

Original entry on oeis.org

2, 5, 13, 17, 29, 37, 41, 53, 73, 89, 137, 149, 157, 181, 257, 269, 281, 293, 313, 349, 373, 397, 401, 409, 421, 461, 541, 557, 577, 593, 661, 709, 733, 757, 769, 773, 797, 853, 937, 953, 1021, 1049, 1069, 1181, 1237, 1277, 1301, 1373, 1429, 1433, 1453, 1489

Offset: 1

Thomas Ordowski, Apr 20 2012

nonn

			13 = 2^2 + 3^2, 2^4 + 3^4 = 97 is prime.

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

Subsequence of A002313.
Cf. A003336 (numbers that are the sum of 2 nonzero 4th powers).
Cf. A002645 (quartan primes: primes of the form x^4 + y^4).

nn = 40; t = {}; Do[c = a^2 + b^2; If[c < nn^2 && PrimeQ[c] && PrimeQ[a^4 + b^4], AppendTo[t, c]], {a, nn}, {b, a}]; Sort[t] (* T. D. Noe, Apr 22 2012 *)
Take[#[[1]]^2+#[[2]]^2&/@Select[Tuples[Range[40],2],AllTrue[{#[[1]]^2+ #[[2]]^2, #[[1]]^4+#[[2]]^4},PrimeQ]&]//Union,60] (* Harvey P. Dale, Jun 25 2018 *)

list(lim)=my(v=List(),t);lim\=1;for(x=1,sqrtint(lim),for(y=1, min(sqrtint(lim-x^2),x), if(isprime(t=x^2+y^2)&&isprime(x^4+y^4), listput(v,t)))); vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Apr 22 2012

A250545 Numbers of the form 4^x + y^4 with x, y >= 0.

Original entry on oeis.org

1, 2, 4, 5, 16, 17, 20, 32, 64, 65, 80, 82, 85, 97, 145, 256, 257, 260, 272, 320, 337, 512, 626, 629, 641, 689, 881, 1024, 1025, 1040, 1105, 1280, 1297, 1300, 1312, 1360, 1552, 1649, 2320, 2402, 2405, 2417, 2465, 2657, 3425, 4096, 4097, 4100

Offset: 1

Vincenzo Librandi, Nov 26 2014

No terms are divisible by 3, 7 or 11.

Subsequence of A001481 and A250482.

			17 is in this sequence because 4^0+2^4 = 4^2+1^4=17.
82 is in this sequence because 4^0+3^4 = 82.

Cf. A001481, A002645.

Cf. similar sequences listed in A250482.

nn=10; Union[Select[Flatten[Table[4^x + y^4, {x, 0, nn}, {y, 0, nn}]], #<=nn^4 &]]

A291206 Semi-octavan primes: primes of the form x^4 + y^8.

Original entry on oeis.org

2, 17, 257, 337, 881, 1297, 2657, 6577, 10657, 14897, 16561, 28817, 65537, 65617, 66161, 80177, 83777, 149057, 160001, 166561, 260017, 280097, 331777, 391921, 394721, 411361, 463537, 596977, 614657, 621217, 847601, 1055137, 1336337, 1342897, 1682017, 1763137

Offset: 1

Charles R Greathouse IV, Aug 21 2017

nonn

			a(1) = 1^4 + 1^8 = 2.
a(2) = 2^4 + 1^8 = 17.
a(3) = 1^4 + 2^8 = 257.
a(4) = 3^4 + 2^8 = 337.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
A. J. C. Cunningham, High quartan factorisations and primes, Messenger of Mathematics 36 (1907), pp. 145-174.

Subsequence of A002645 and hence of A028916. A006686 is a subsequence.

Take[Select[Flatten[Table[x^4+y^8,{x,40},{y,40}]],PrimeQ]//Union,40] (* Harvey P. Dale, May 01 2025 *)

list(lim)=my(v=List([2]),x4,t); for(x=1, sqrtnint(lim\=1,4), x4=x^4; forstep(y=x%2+1, sqrtnint(lim-x4,8), 2, if(isprime(t=x4+y^8), listput(v, t)))); Set(v)

A182277 Quartan semiprimes: semiprimes of the form x^4 + y^4, x>0, y>0.

Original entry on oeis.org

82, 626, 706, 1921, 2402, 4097, 6497, 6817, 7186, 8962, 10001, 10081, 14642, 17042, 18737, 20737, 21202, 21361, 23137, 24641, 28562, 28642, 29186, 29857, 35377, 38417, 38497, 43202, 44977, 50641, 53026, 53057, 65266, 67937, 72097, 83522, 83602, 84146, 84817, 85922

Offset: 1

Jonathan Vos Post, Apr 22 2012

nonn

This is to A002645 as A001358 semiprimes is to A000040 primes.

			a(1) = 3^4 + 1^4 = 82 = 2 * 41.

George Greaves, On the representation of a number as a sum of two fourth powers, MATHEMATISCHE ZEITSCHRIFT, Volume 94, Number 3 (1966), 223-234, DOI: 10.1007/BF01111351.

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

Cf. A003336 Numbers that are the sum of 2 nonzero 4th powers, A002645 Quartan primes: primes of the form x^4 + y^4, x>0, y>0.

issemi(n)=bigomega(n)==2
list(lim)=my(v=List(),t);for(x=1,(lim+.5)^(1/4),for(y=1,min(x,(lim-x^4 + .5)^(1/4)),if(issemi(t=x^4+y^4),listput(v,t))));vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Apr 22 2012

A001358 INTERSECTION A003336.

a(12)-a(40) from Charles R Greathouse IV, Apr 22 2012

A250717 Primes of the form 4^x + y^4 with x, y > 0.

Original entry on oeis.org

5, 17, 97, 257, 337, 641, 881, 2417, 2657, 4177, 4721, 6577, 10657, 14657, 14897, 28817, 54721, 65537, 65617, 66161, 80177, 83537, 83777, 130337, 134417, 149057, 260017, 279857, 280097, 283937, 394721, 531457, 596977, 1049201, 1050977, 1055137, 1178897

Offset: 1

Vincenzo Librandi, Nov 27 2014

Subsequence of A250481. Also, except for the first term, subsequence of A002645.

			257 is in this sequence because 4^4+1^4=257.
881 is in this sequence because 4^4+5^4=881.

Cf. A250545.

Cf. similar sequences listed in A250481.

f[x_, y_]:= 4^x + y^4; lst={}; Do[p=f[x, y]; If[PrimeQ[p], AppendTo[lst, p]], {y, 50}, {x, 50}]; Take[Union[lst], 50]

A282867 Primes of the form x^2 + y^2 with x > y such that x^2 - y^2 is a square and x^4 + y^4 is a prime.

Original entry on oeis.org

41, 313, 3593, 4481, 32633, 42961, 66361, 67073, 165233, 198593, 237161, 266921, 378953, 462073, 465041, 487073, 559001, 594161, 750353, 757633, 815401, 1157033, 1414081, 1416161, 1687393, 2439881, 2793481, 2866121, 2947561, 3344161, 3577913, 3759713, 4295281, 4617073, 4795481, 5654641

Offset: 1

Thomas Ordowski and Altug Alkan, Feb 23 2017

nonn

Primes of the form (u^4 + v^4)/2 with u and v odd and (u^8 + 6*u^4*v^4 + v^8)/8 prime. - Robert Israel, Feb 24 2017

			For prime 41 = 5^2 + 4^2 is 5^2 - 4^2 = 3^2 and 5^4 + 4^4 = 881 is prime.

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

Subsequence of A002646.

Cf. A002144, A002645.

N:= 10^7: # to get all terms <= N Res:= {}:
for w from 1 to floor((2*N)^(1/4)) by 2 do
  for u from 1 to min(w-1, floor((2*N-w^4)^(1/4))) by 2 do
    p:= (u^4 + w^4)/2;
    if isprime(p) and isprime((u^8 + 6*u^4*w^4 + w^8)/8) then
      Res:= Res union {p}
    fi;
od od:
sort(convert(Res,list)); # Robert Israel, Feb 24 2017

Select[Total[#^2]&/@Select[Subsets[Range[3000],{2}],IntegerQ[Sqrt[#[[2]]^2-#[[1]]^2]] && PrimeQ[ Total[#^4]]&],PrimeQ]//Union (* Harvey P. Dale, Jul 23 2024 *)

a(n) == 1 (mod 8).

a(n) == 1 or 33 (mod 40).

cp's OEIS Frontend

A233567 Number of ways to write n = p + q (q > 0) with p and p^4 + phi(q)^4 both prime, where phi(.) is Euler's totient function (A000010).

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A100271 Primes of the form a^4 + b^3 with b>0.

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PARI

A216285 Primes which cannot be written as x^4+y^4.

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Formula

A247857 Primes of the form a^2 + b^4, with repetition.

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A182198 Primes of form a^2 + b^2 such that a^4 + b^4 is prime.

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PARI

A250545 Numbers of the form 4^x + y^4 with x, y >= 0.

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Mathematica

A291206 Semi-octavan primes: primes of the form x^4 + y^8.

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PARI

A182277 Quartan semiprimes: semiprimes of the form x^4 + y^4, x>0, y>0.

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PARI