A100270 -id:A100270 - cp's OEIS frontend

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

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.

A100268 Primes of the form x^4 + y^4 with x^2 + y^2 and x+y also prime.

Original entry on oeis.org

2, 17, 97, 257, 641, 1297, 4177, 4721, 12401, 15937, 16561, 38561, 65537, 83537, 89041, 105601, 140321, 160081, 204481, 283937, 284881, 384817, 391921, 411361, 462097, 471617, 531457, 643217, 824641, 838561, 1049201, 1089841, 1342897

Offset: 1

T. D. Noe, Nov 11 2004

nonn

The first Mathematica program generates numbers of the form x^4 + y^4 in order of increasing magnitude; it accepts a number when all the x^2^k + y^2^k are prime for k=0,1,2.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Generalized Fermat Number

Cf. A099332, A100269, A100270.

n=2; 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]]; allPrime=True; k=0; While[k<=n&&allPrime, allPrime=PrimeQ[x^2^k+y^2^k]; k++ ]; !allPrime]; p, {40}]
With[{nn=40},Select[Union[Transpose[Select[Total/@{#^4,#^2,#}&/@ Tuples[ Range[nn],2],AllTrue[#,PrimeQ]&]][[1]]],#<=nn^4+1&]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 23 2015 *)

A100269 Primes of the form x^8 + y^8 with x^4 + y^4, x^2 + y^2 and x+y also prime.

Original entry on oeis.org

2, 257, 65537, 2724909545357921, 3282116715437377, 40213879071634241, 147578912575757441, 303879829574456257, 697576026529536481, 1316565220482548321, 2860283484326400961, 4080251077774711937

Offset: 1

T. D. Noe, Nov 11 2004

nonn

The Mathematica program generates numbers of the form x^8 + y^8 in order of increasing magnitude; it accepts a number when all the x^2^k + y^2^k are prime for k=0,1,2,3.

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, Generalized Fermat Number.

Cf. A099332, A100268, A100270.

n=3; 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]]; allPrime=True; k=0; While[k<=n&&allPrime, allPrime=PrimeQ[x^2^k+y^2^k]; k++ ]; !allPrime]; p, {20}]

A164307 Primes in A081175.

Original entry on oeis.org

3, 5, 17, 257, 65537

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 12 2009

nonn

The 6th term is too large to include in the data section (see Example section or the b-file).
Primes of the form sum_{j=1..u} j^x for some x>0, u>1. (Since the case of x=1 leads to the triangular numbers with no additional primes, this is equivalent to the definition.)
Primes in A000330 (x=2), or in A000537 (x=3), or in A000538 (x=4), or in A000539 (x=5) etc. See A164312 for the corresponding x values.

			a(1) = 1^1 + 2^1 = 3.
a(2) = 1^2 + 2^2 = 5.
a(3) = 1^4 + 2^4 = 17.
a(4) = 1^8 + 2^8 = 257.
a(5) = 1^16 + 2^16 = 65537.
a(6) = 1^1440 + 2^1440 + 3^1440 + 4^1440 + 5^1440 = 3.287049497374559048967261852*10^1006 = 3287049497374559048967261852 ... 458593539025033893379.

N. J. A. Sloane, Table of n, a(n) for n = 1..6

Cf. A000215, A070592, A019434, A092506, A093179, A100270, A123599.

lst={};Do[s=0;Do[If[PrimeQ[s+=n^x],AppendTo[lst,s];Print[Date[],s]],{n, 4!}],{x,7!}];lst

Edited by R. J. Mathar, Aug 22 2009

Corrected by N. J. A. Sloane, Nov 23 2015 at the suggestion of Jaroslav Krizek.

A164312 Numbers n such that k^n + (k-1)^n + ... + 3^n + 2^n + 1 is prime for some k.

Original entry on oeis.org

1, 2, 4, 8, 16, 1440

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 12 2009

These terms have k-values {2, 2, 2, 2, 2, 5} respectively. When k = 2, the prime mentioned in the definition is given in A164307. - Derek Orr, Jun 06 2014

			1^1 + 2^1 = 3 is prime (k = 2).
1^2 + 2^2 = 5 is prime (k = 2).
1^4 + 2^4 = 17 is prime (k = 2).
1^8 + 2^8 = 257 is prime (k = 2).
1^16 + 2^16 = 65537 is prime (k = 2).
1^1440 + 2^1440 + 3^1440 + 4^1440 + 5^1440 = 3.287049497374559048967261852*10^1006 = 3287049497374559048967261852 ... 458593539025033893379 is prime (k = 5).

Cf. A000215, A070592, A019434, A092506, A093179, A100270, A123599, A164307.

lst={};Do[s=0;Do[If[PrimeQ[s+=n^x],AppendTo[lst,x];Print[Date[],x]],{n,4!}],{x,7!}];lst

a(n)=for(k=1,10^3,if(ispseudoprime(sum(i=1,k,i^n)),return(k)))
n=1;while(n<5000,if(a(n),print1(n,", "));n++) \\ Derek Orr, Jun 06 2014

Definition improved by Derek Orr, Jun 06 2014

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

A100268 Primes of the form x^4 + y^4 with x^2 + y^2 and x+y also prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A100269 Primes of the form x^8 + y^8 with x^4 + y^4, x^2 + y^2 and x+y also prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A164307 Primes in A081175.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A164312 Numbers n such that k^n + (k-1)^n + ... + 3^n + 2^n + 1 is prime for some k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions