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.

Showing 1-4 of 4 results.

A181141 Numbers k such that (8^k + 7^k)/15 is prime.

Original entry on oeis.org

7, 13, 31, 43, 269, 353, 383, 619, 829, 877, 4957, 5711, 8317, 21739, 24029, 38299
Offset: 1

Views

Author

Jean-Louis Charton, Oct 06 2010

Keywords

Comments

Terms 21739 and 24029 were found by Lelio R. Paula in February and April 2009. Term 38299 was found by Jean-Louis Charton in October 2010 corresponding to a probable prime with 34587 digits.
a(17) > 10^5. - Robert Price, Dec 29 2012

Links

Crossrefs

See also A062574.

Programs

A059803 Numbers n such that 9^n - 8^n is prime or a strong pseudoprime.

Original entry on oeis.org

2, 7, 29, 31, 67, 149, 401, 2531, 19913, 30773, 53857, 170099
Offset: 1

Views

Author

Mike Oakes, Feb 23 2001

Keywords

Comments

Some of the larger entries may only correspond to probable primes.
In general, for any positive integers n, a and b, a>b, a necessary condition for a^n-b^n to be prime is that either a-b=1 and n be a prime or n=1 and a-b be prime (from Arturo Magidin and Hagman in Sci.Math, Sep 11, 2010). - Vincenzo Librandi, Sep 12 2010

Crossrefs

Cf. A000043, A057468, A059801, A059802, A062572-A062574, A062576-A062666.

Programs

Extensions

Three more terms found by Jean-Louis Charton in 2004-2005: a(9) = 30773, a(10) = 53857, a(11) = 170099. - Alexander Adamchuk, Dec 08 2006

A247093 Triangle read by rows: T(m,n) = smallest odd prime p such that (m^p-n^p)/(m-n) is prime (0

Original entry on oeis.org

3, 3, 3, 0, 0, 3, 3, 5, 13, 3, 3, 0, 0, 0, 5, 5, 3, 3, 5, 3, 3, 3, 0, 3, 0, 19, 0, 7, 0, 3, 0, 0, 3, 0, 3, 7, 19, 0, 3, 0, 0, 0, 31, 0, 3, 17, 5, 3, 3, 5, 3, 5, 7, 5, 3, 3, 0, 0, 0, 3, 0, 3, 0, 0, 0, 3, 5, 3, 7, 5, 5, 3, 7, 3, 3, 251, 3, 17, 3, 0, 5, 0, 151, 0, 0, 0, 59, 0, 5, 0, 3, 3, 5, 0, 1097, 0, 0, 3, 3, 0, 0, 7, 0, 17, 3
Offset: 1

Views

Author

Eric Chen, Nov 18 2014

Keywords

Comments

T(m,n) is 0 if and only if m and n are not coprime or A052409(m) and A052409(n) are not coprime. (The latter has some exceptions, like T(8,1) = 3. In fact, if p is a prime and does not equal to A052410(gcd(A052409(m),A052409(n))), then (m^p-n^p)/(m-n) is composite, so if it is not 0, then it is A052410(gcd(A052409(m),A052409(n))).) - Eric Chen, Nov 26 2014
a(i) = T(m,n) corresponds only to probable primes for (m,n) = {(15,4), (18,1), (19,18), (31,6), (37,22), (37,25), ...} (i={95, 137, 171, 441, 652, 655, ...}). With the exception of these six (m,n), all corresponding primes up to a(663) are definite primes. - Eric Chen, Nov 26 2014
a(n) is currently known up to n = 663, a(664) = T(37, 34) > 10000. - Eric Chen, Jun 01 2015
For n up to 1000, a(n) is currently unknown only for n = 664, 760, and 868. - Eric Chen, Jun 01 2015

Examples

			Read by rows:
m\n        1   2   3   4   5   6   7   8   9   10  11
2          3
3          3   3
4          0   0   3
5          3   5   13  3
6          3   0   0   0   5
7          5   3   3   5   3   3
8          3   0   3   0   19  0   7
9          0   3   0   0   3   0   3   7
10         19  0   3   0   0   0   31  0   3
11         17  5   3   3   5   3   5   7   5   3
12         3   0   0   0   3   0   3   0   0   0   3
etc.

Links

Crossrefs

Cf. A128164 (n,1), A125713 (n+1,n), A125954 (2n+1,2), A122478 (2n+1,2n-1).
Cf. A000043 (2,1), A028491 (3,1), A057468 (3,2), A059801 (4,3), A004061 (5,1), A082182 (5,2), A121877 (5,3), A059802 (5,4), A004062 (6,1), A062572 (6,5), A004063 (7,1), A215487 (7,2), A128024 (7,3), A213073 (7,4), A128344 (7,5), A062573 (7,6), A128025 (8,3), A128345 (8,5), A062574 (8,7), A173718 (9,2), A128346 (9,5), A059803 (9,8), A004023 (10,1), A128026 (10,3), A062576 (10,9), A005808 (11,1), A210506 (11,2), A128027 (11,3), A216181 (11,4), A128347 (11,5), A062577 (11,10), A004064 (12,1), A128348 (12,5), A062578 (12,11).

Programs

  • Mathematica
    t1[n_] := Floor[3/2 + Sqrt[2*n]]
m[n_] := Floor[(-1 + Sqrt[8*n-7])/2]
t2[n_] := n-m[n]*(m[n]+1)/2
b[n_] := GCD @@ Last /@ FactorInteger[n]
is[m_, n_] := GCD[m, n] == 1 && GCD[b[m], b[n]] == 1
Do[k=2, If[is[t1[n], t2[n]], While[ !PrimeQ[t1[n]^Prime[k] - t2[n]^Prime[k]], k++]; Print[Prime[k]], Print[0]], {n, 1, 663}] (* Eric Chen, Jun 01 2015 *)
  • PARI
    a052409(n) = my(k=ispower(n)); if(k, k, n>1);
a(m, n) = {if (gcd(m,n) != 1, return (0)); if (gcd(a052409(m), a052409(n)) != 1, return (0)); forprime(p=3,, if (isprime((m^p-n^p)/(m-n)), return (p)););}
tabl(nn) = {for (m=2, nn, for(n=1, m-1, print1(a(m,n), ", ");); print(););} \\ Michel Marcus, Nov 19 2014
  • PARI
    t1(n)=floor(3/2+sqrt(2*n))
t2(n)=n-binomial(floor(1/2+sqrt(2*n)), 2)
b(n)=my(k=ispower(n)); if(k, k, n>1)
a(n)=if(gcd(t1(n),t2(n)) !=1 || gcd(b(t1(n)), b(t2(n))) !=1, 0, forprime(p=3,2^24,if(ispseudoprime((t1(n)^p-t2(n)^p)/(t1(n)-t2(n))), return(p)))) \\ Eric Chen, Jun 01 2015

A147670 Primes of the form 8^k-7^k.

Original entry on oeis.org

1273609, 6612607849, 2019169299698041, 151522599154859354635552921, 9745744932248196392577951049, 220850092209355591999793321040563526382178219220559368038951319096656329
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Nov 10 2008

Keywords

Comments

The next term (a(7)) has 103 digits. - Harvey P. Dale, Jul 05 2023

Examples

			8^7-7^7 = 2097152-823543 = 1273609.

Crossrefs

Cf. A016177, A062574.

Programs

  • Mathematica
    lst={};Do[p=8^n-7^n;If[PrimeQ[p],AppendTo[lst,p]],{n,6!}];lst
Select[Table[8^n-7^n,{n,100}],PrimeQ] (* Harvey P. Dale, Jul 05 2023 *)
Showing 1-4 of 4 results.

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