A006686 -id:A006686 - cp's OEIS frontend

A331524 a(n) is the least positive k such that A006686(n) - k^8 is an eighth power.

1, 1, 1, 5, 3, 7, 2, 8, 3, 8, 9, 13, 4, 4, 17, 7, 17, 2, 8, 3, 5, 20, 7, 11, 17, 8, 23, 19, 10, 20, 3, 11, 17, 10, 22, 19, 14, 22, 17, 21, 25, 11, 13, 23, 27, 8, 32, 25, 33, 6, 25, 35, 7, 23, 31, 10, 37, 16, 18, 39, 5, 7, 42, 41, 30, 36, 5, 11, 8, 18, 30, 36

Offset: 1

Rémy Sigrist, Jan 19 2020

nonn

			The first terms, alongside A006686(n), are:
  n   a(n)  A006686(n)
  --  ----  -----------------------
   1     1           2 = 1^8 +  1^8
   2     1         257 = 1^8 +  2^8
   3     1       65537 = 1^8 +  4^8
   4     5     2070241 = 5^8 +  6^8
   5     3   100006561 = 3^8 + 10^8
   6     7   435746497 = 7^8 + 12^8
   7     2   815730977 = 2^8 + 13^8
   8     8   832507937 = 8^8 + 13^8
   9     3  1475795617 = 3^8 + 14^8
  10     8  2579667841 = 8^8 + 15^8

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A331524

See A331435 for similar sequences.

Cf. A001016, A006686.

PARI
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A085316 Duplicate of A006686.

Original entry on oeis.org

2, 257, 65537, 2070241, 100006561, 435746497, 815730977, 832507937, 1475795617

Offset: 1

dead

A100266 Primes of the form x^16 + y^16.

Original entry on oeis.org

2, 65537, 4338014017, 2973697798081, 36054040477057, 314707907280257, 184884411482927041, 665698084159890497, 675416609183179841, 2177953490397261761, 8746361693522261761, 18492693803573123777

Offset: 1

T. D. Noe, Nov 11 2004

nonn

The Mathematica program generates numbers of the form x^16 + y^16 in order of increasing magnitude; it accepts a number when it is prime.

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Generalized Fermat Number.

Cf. A100267 (primes of the form x^32 + y^32), A006686 (primes of the form x^8 + y^8), A002645 (primes of the form x^4 + y^4), A002313 (primes of the form x^2 + y^2).

n=4; pwr=2^n; xmax=2; r=Range[xmax]; num=r^pwr+r^pwr; Table[While[p=Min[num]; x=Position[num, p][[1, 1]]; y=r[[x]]; r[[x]]++; num[[x]]=x^pwr+r[[x]]^pwr; If[x==xmax, xmax++; AppendTo[r, xmax+1]; AppendTo[num, xmax^pwr+(xmax+1)^pwr]]; !PrimeQ[p]]; p, {15}]
q=16;lst={};Do[Do[p=n^q+m^q;If[PrimeQ[p],AppendTo[lst,p]],{n,0,5!}],{m,0,5!}];lst;Length[lst];Take[Union[lst],55] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2009 *)
Union[Select[Total[#^16]&/@Tuples[Range[20],2],PrimeQ]] (* Harvey P. Dale, Nov 03 2013 *)

A100267 Primes of the form x^32 + y^32.

Original entry on oeis.org

2, 3512911982806776822251393039617, 2211377674535255285545615254209921, 476961452964007550415682034114910337, 14748002492224459115975467901357427939457

Offset: 1

T. D. Noe, Nov 11 2004

nonn

The Mathematica program generates numbers of the form x^32 + y^32 in order of increasing magnitude; it accepts a number when it is prime.

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Generalized Fermat Number.

Cf. A100266 (primes of the form x^16 + y^16), A006686 (primes of the form x^8 + y^8), A002645 (primes of the form x^4 + y^4), A002313 (primes of the form x^2 + y^2).

n=5; pwr=2^n; xmax=2; r=Range[xmax]; num=r^pwr+r^pwr; Table[While[p=Min[num]; x=Position[num, p][[1, 1]]; y=r[[x]]; r[[x]]++; num[[x]]=x^pwr+r[[x]]^pwr; If[x==xmax, xmax++; AppendTo[r, xmax+1]; AppendTo[num, xmax^pwr+(xmax+1)^pwr]]; !PrimeQ[p]]; p, {10}]

A111635 Smallest prime of the form x^(2^n) + y^(2^n) where x,y are distinct integers.

Original entry on oeis.org

2, 5, 17, 257, 65537, 3512911982806776822251393039617, 4457915690803004131256192897205630962697827851093882159977969339137, 1638935311392320153195136107636665419978585455388636669548298482694235538906271958706896595665141002450684974003603106305516970574177405212679151205373697500164072550932748470956551681

Offset: 0

Max Alekseyev, Aug 09 2005

nonn

Is this sequence defined for all n?
From Jeppe Stig Nielsen, Sep 16 2015: (Start)
Numbers of this form are sometimes called extended generalized Fermat numbers.
If we restrict ourselves to the case y=1, we get instead the sequence A123599, therefore a(n) <= A123599(n) for all n. Can this be an equality for some n > 4?
The formula x^(2^m) + y^(2^m) also gives the decreasing chain {A000040, A002313, A002645, A006686, A100266, A100267, ...} of subsets of the prime numbers if we drop the requirement that x != y and take all primes (not just the smallest one) with m greater than some lower bound.
(End)
For more terms (the values of max(x,y)), see A291944. - Jeppe Stig Nielsen, Dec 28 2019

Jeppe Stig Nielsen, Table of n, a(n) for n = 0..9

Cf. A019434, A100270, A123599, A291944.

A283019 Primes which are the sum of three nonzero 8th powers.

Original entry on oeis.org

3, 6563, 72353, 137633, 787811, 1745153, 7444673, 44726593, 49202147, 61503553, 86093443, 91858243, 100006817, 100072097, 101686177, 107444417, 143046977, 200006561, 214756067, 257412163, 300452323, 430372577, 431661313, 435812033, 447149537, 452523713, 489805633, 530372321, 744340577

Offset: 1

Ilya Gutkovskiy, Feb 26 2017

nonn

Primes of form x^8 + y^8 + z^8 where x, y, z > 0.

			3 = 1^8 + 1^8 + 1^8;
6563 = 1^8 + 1^8 + 3^8;
72353 = 2^8 + 3^8 + 4^8, etc.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A001016, A003381, A006686, A007490, A085317, A085318, A085319, A283017, A283018.

nn = 13; Select[Union[Plus @@@ (Tuples[Range[nn], {3}]^8)], # <= nn^8 && PrimeQ[#] &]

list(lim)=my(v=List(),A,B,t); lim\=1; for(a=1,sqrtnint(lim-2,8), A=a^8; for(b=1,min(sqrtnint(lim-A-1,8),a), B=A+b^8; forstep(c=if(B%2,2,1),sqrtnint(lim-B,8),2, if(isprime(t=B+c^8), listput(v,t))))); Set(v) \\ Charles R Greathouse IV, Nov 05 2017

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)

A290780 Half-octavan primes: primes of the form (x^8 + y^8)/2.

Original entry on oeis.org

198593, 21523361, 107182721, 407865361, 429388721, 3487882001, 11979660241, 39155495921, 84785726833, 141217650641, 141321947681, 250123401793, 253611085201, 289278699121, 391337974721, 426445714033, 426448401121

Offset: 1

Charles R Greathouse IV, Aug 20 2017

nonn

			a(1) = (5^8 + 3^8)/2 = 198593.
a(2) = (9^8 + 1^8)/2 = 21523361.
a(3) = (11^8 + 3^8)/2 = 107182721.
a(4) = (13^8 + 1^8)/2 = 407865361.
a(5) = (13^8 + 9^8)/2 = 429388721.

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.

Cf. A006686, A002646.

N:= 10^12: # to get all terms <= N
sort(convert(select(isprime, {seq(seq((x^8+y^8)/2, y= (x mod 2)..min(x,floor((2*N-x^8)^(1/8))),2),x=1..floor((2*N)^(1/8)))}),list)); # Robert Israel, Aug 21 2017

Sort[Select[Total/@(Union[Sort/@Tuples[Range[0, 50], 2]]^8)/2, PrimeQ]] (* or *) lst={};Do[If[PrimeQ[(a^8 + b^8) / 2], AppendTo[lst, (a^8 + b^8) / 2]], {a, 100}, {b, a, 100}]; Sort[lst] (* Vincenzo Librandi, Aug 21 2017 *)

list(lim)=my(v=List(),x8,t); forstep(x=1,sqrtnint(lim\=1,8),2, x8=x^8; forstep(y=1,min(sqrtnint(lim-x8,8), x-1),2, if(isprime(t=(x8+y^8)/2), listput(v,t)))); Set(v) \\ Charles R Greathouse IV, Aug 20 2017

cp's OEIS Frontend

A331524 a(n) is the least positive k such that A006686(n) - k^8 is an eighth power.

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PARI

A085316 Duplicate of A006686.

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A100266 Primes of the form x^16 + y^16.

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Mathematica

A100267 Primes of the form x^32 + y^32.

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Mathematica

A111635 Smallest prime of the form x^(2^n) + y^(2^n) where x,y are distinct integers.

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A283019 Primes which are the sum of three nonzero 8th powers.

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Mathematica

PARI

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

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Mathematica

PARI

A290780 Half-octavan primes: primes of the form (x^8 + y^8)/2.

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Mathematica

PARI