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.

A130060 Primes p such that p^2 divides 3^p - 2^p - 1; or primes in A127074(n).

Original entry on oeis.org

2, 3, 7, 179, 619, 17807
Offset: 1

Views

Author

Alexander Adamchuk, May 05 2007

Keywords

Comments

The prime p divides 3^p - 2^p - 1. Quotients (3^p - 2^p - 1)/p, where p = Prime[n], are listed in A127071. - Alexander Adamchuk, Jan 31 2008
a(7) > 10^9. [From D. S. McNeil, Mar 16 2009]

Crossrefs

Cf. A127071, A127072, A127073, A127074 = numbers n such that n^2 divides 3^n - 2^n - 1. Cf. A130058, A130059, A130061, A130062, A130063.

Programs

  • Mathematica
    Do[ n=Prime[k]; f=PowerMod[3,n,n^2] - PowerMod[2,n,n^2] - 1; If[ IntegerQ[ f/n^2 ], Print[n] ], {k,1,100000} ]

Extensions

2 more terms found by Ryan Propper, Jan 01 2008.
Incorrect a(7), a(8) removed by D. S. McNeil, Mar 16 2009. (The old version was 2,3,7,179,619,17807,135433,1376257.)

A130061 Numbers k that divide 3^((k-1)/2) - 2^((k-1)/2) - 1.

Original entry on oeis.org

1, 3, 35, 147, 195, 219, 291, 399, 579, 583, 723, 939, 1011, 1023, 1227, 1299, 1371, 1443, 1731, 1803, 2019, 2307, 2499, 2811, 3003, 3027, 3099, 3387, 3459, 3603, 3747, 3891, 3963, 4467, 4623, 4827, 4971, 5187, 5259, 5331, 5403, 5619, 5979, 6051, 6267
Offset: 1

Views

Author

Alexander Adamchuk, May 05 2007

Keywords

Comments

It appears that all terms are composite except a(1) = 1 and a(2) = 3. Most listed terms are divisible by 3, except {1, 35, 583, 70643, ...}.

Links

Crossrefs

Cf. A097934 (primes p that divide 3^((p-1)/2) - 2^((p-1)/2)).
Cf. A038876 (primes p such that 6 is a square mod p).
Cf. A127071, A127072, A127073, A127074.
Cf. A130058, A130059, A130060, A130062, A130063.

Programs

  • Mathematica
    Select[ Range[10000], Mod[ PowerMod[3,(#-1)/2,# ] - PowerMod[2,(#-1)/2,# ] -1, # ]==0&]

A130063 Primes p such that p divides 3^((p+1)/2) - 2^((p+1)/2) - 1.

Original entry on oeis.org

23, 47, 71, 73, 97, 167, 191, 193, 239, 241, 263, 311, 313, 337, 359, 383, 409, 431, 433, 457, 479, 503, 577, 599, 601, 647, 673, 719, 743, 769, 839, 863, 887, 911, 937, 983, 1009, 1031, 1033, 1103, 1129, 1151, 1153, 1201, 1223, 1249, 1297, 1319, 1321, 1367
Offset: 1

Views

Author

Alexander Adamchuk, May 05 2007

Keywords

Comments

Primes = 1 or 23 mod 24. Hence, together with 2, primes such that (2/p) = 1 = (3/p) where (k/p) is the Legendre symbol. - Charles R Greathouse IV, Apr 06 2012

Links

Crossrefs

Cf. A097934 = Primes p such that p divides 3^((p-1)/2) - 2^((p-1)/2).
Subsequence of A038876.
Cf. A127071, A127072, A127073, A127074. Cf. A130058, A130059, A130060, A130061, A130062.

Programs

  • Mathematica
    Select[ Range[2000], PrimeQ[ # ]&&Mod[ PowerMod[3,(#+1)/2,# ] - PowerMod[2,(#+1)/2,# ] - 1, # ]==0&]
Select[Prime[Range[250]],Divisible[3^((#+1)/2)-2^((#+1)/2)-1,#]&] (* Harvey P. Dale, Mar 21 2021 *)
  • PARI
    is(n)=(n+1)%24<3 && isprime(n) \\ Charles R Greathouse IV, Apr 06 2012

A260507 Primes p such that (2^p+1)^(p-1) == 1 (mod p^2).

Original entry on oeis.org

2, 7, 179, 619, 17807
Offset: 1

Views

Author

Felix Fröhlich, Jul 27 2015

Keywords

Comments

A000040(n) such that A260531(n) = 1.
Is this a subsequence of A130060?
a(6) > 10325801 if it exists.
a(6) > 3037000499 if it exists. - Hiroaki Yamanouchi, Aug 20 2015

Examples

			2^7 + 1 = 129 and 129^6 == 1 (mod 7^2), so 7 is a term of the sequence.

Crossrefs

Cf. A098640, A127074, A130060, A130062, A260531.

Programs

  • Mathematica
    Select[Prime@ Range@ 120, Mod[(2^# + 1)^(# - 1), #^2] == 1 &] (* Michael De Vlieger, Jul 29 2015 *)
  • PARI
    forprime(p=2, , if(Mod(2^p+1, p^2)^(p-1)==1, print1(p, ", ")))
Showing 1-4 of 4 results.

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