cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A257917 a(n) is the largest y that is a member of a pair (x, y) of integers with x - y > 1 such that x^2 - y^2 is equal to the Fermat number 2^(2^n) + 1, or 0 if no such number exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 3349888, 33640210518272, 2852314775548000778752, 46730819857678988884581779099803448292025618770199631109363712
Offset: 0

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Author

Arkadiusz Wesolowski, May 12 2015

Keywords

Comments

2^(2^n) + 1 belongs to A019434 if and only if a(n) = 0.

References

  • M. Krizek, F. Luca, L. Somer, 17 Lectures on Fermat Numbers: From Number Theory to Geometry, CMS Books in Mathematics, vol. 9, Springer-Verlag, New York, 2001, p. 6.

Links

Crossrefs

Cf. A000215, A257916.

Programs

  • PARI
    a(n) = {my(fn = 2^(2^n) + 1); if (isprime(fn), return(0)); my(spf = factor(fn)[1,1]); (fn/spf - spf)/2;} \\ Michel Marcus, Jun 07 2015

Formula

If 2^(2^n) + 1 is composite, then a(n) = A257916(n) - A093179(n).

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

Views

Author

Vladimir Joseph Stephan Orlovsky, Aug 12 2009

Keywords

Comments

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

Examples

			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).

Crossrefs

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

Programs

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

Extensions

Definition improved by Derek Orr, Jun 06 2014

A358684 a(n) is the minimum integer k such that the smallest prime factor of the n-th Fermat number exceeds 2^(2^n - k).

Original entry on oeis.org

0, 0, 0, 0, 0, 23, 46, 73, 206, 491, 999, 2030, 4080, 8151
Offset: 0

Views

Author

Lorenzo Sauras Altuzarra, Nov 26 2022

Keywords

Comments

2^(2^n - a(n)) < A093179(n).
Conjecture: the dyadic valuation of A093179(n) - 1 does not exceed 2^n - a(n).
a(14) is probably equal to 16208; a(15) to a(19) are 32738, 65507, 131028, 262121, 524252; a(20) is unknown; a(21) to a(23) are 2097110, 4194189, 8388581; a(24) is unknown.

Examples

			For n=5, the smallest prime factor of F(5) = 2^(2^5) + 1 is 641 and it falls between 2^(2^5 - 23) = 512 < 641 < 1024 = 2^(2^5 - 22) so that a(5) = 23.

Links

Crossrefs

Cf. A000215, A093179.

Formula

Conjecture: a(n) ~ 2^n as n -> oo.

A372867 Distinct terms in A242017, listed in the order of their appearance.

Original entry on oeis.org

3, 5, 17, 97, 641, 257, 193, 274177, 65537, 449, 59649589127497217, 769, 1238926361552897, 5441, 5953, 2424833, 7873, 2753, 3329, 10753, 45592577, 18433, 4673, 15937, 444929, 11777, 12161, 21698561, 6977
Offset: 1

Views

Author

Jean-Marc Rebert, May 15 2024

Keywords

Comments

Conjecture: every term except 3 belongs to A366609. - Bill McEachen, Jun 12 2024

Crossrefs

Cf. A023394, A070592, A093179, A242017.
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