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 1-3 of 3 results.

A175932 Smallest prime p such that there exist exactly n integers b such that 1 < b < p and b^(p-1) == 1 (mod p^2) or, equivalently, Fermat quotient q_p(b) == 0 (mod p).

Original entry on oeis.org

2, 29, 11, 269, 487, 653, 5107, 103291, 40487, 2544079, 1093, 3511, 1006003
Offset: 0

Views

Author

Max Alekseyev, Oct 24 2010

Keywords

Comments

a(n) is the smallest prime p such that A242830(PrimePi(p)) = n, PrimePi = A000720. - Jianing Song, Jan 27 2019

Examples

			a(5) = 653 since 653 is the smallest prime with exactly five bases b = 84, 120, 197, 287, 410.

Links

Crossrefs

Cf. A001220, A130912, A248865, A242830.
Subset of A134307.
Cf. also A255203, A255204, A255205, A255206, A255207, A255208, A255209, A255210.

Programs

  • PARI
    first_n_entries(n)=v=vector(n); toGo=n; forprime(p=2, , count=sum(b=2, p-1, Mod(b, p^2)^(p-1)==1); if(count<=(n-1)&!v[count+1], v[count+1]=p; toGo--; if(!toGo, return(v)))) \\ Jeppe Stig Nielsen, Jul 31 2015, changed to include a(0) = 2 by Jianing Song, Feb 05 2019

Extensions

a(0) = 2 prepended by Jianing Song, Jan 27 2019

A178844 First nonzero Fermat quotient mod the n-th prime.

Original entry on oeis.org

1, 1, 3, 2, 5, 3, 13, 3, 17, 1, 6, 1, 23, 25, 44, 36, 8, 36, 10, 2, 56, 19, 48, 6, 57, 92, 59, 13, 67, 83, 18, 17, 53, 30, 96, 56, 82, 67, 47, 3, 50, 148, 50, 104, 175, 135, 109, 189, 201, 68, 7, 26, 142, 247, 225, 128, 260, 109, 70, 74, 58, 78, 294, 175, 120, 175, 139, 153
Offset: 1

Views

Author

Jonathan Sondow, Jun 24 2010

Keywords

Comments

First nonzero value of q_p(m) mod p with gcd(m,p) = 1, where q_p(m) = (m^(p-1) - 1)/p is the Fermat quotient of p to the base m and p is the n-th prime p_n.
It is believed that a(n) = q_p(3) mod p, if p = p_n is a Wieferich prime A001220. See Section 1.1 in Ostafe-Shparlinski (2010).
See additional comments, references, links, and cross-refs in A001220 and A178815.

Examples

			p_1 = 2 and (m^1 - 1)/2 = 0, 1 == 0, 1 (mod 2) for m = 1, 3, so a(1) = 1.
p_5 = 11 and (m^10 - 1)/11 = 0, 93 == 0, 5 (mod 7) for m = 1, 2, so a(4) = 5.
p_183 = 1093 and (m^1092 - 1)/1093 == 0, 0, 312 (mod 1093) for m = 1, 2, 3, so a(183) = 312.
Similarly, a(490) = 7.

Links

Crossrefs

Cf. A001220, A130912, A178815.

Formula

a(n) = q_p(A178815(n)) mod p, where p = p_n.
a(n) = A130912(n), if n > 1 and p_n is not a Wieferich prime. (Note: the offset of A130912 is n = 2.)

Extensions

Nonexistent A-numbers removed by Jonathan Sondow, Jun 26 2010

A128465 Numbers k such that k divides the numerator of alternating Harmonic number H'((k+1)/2) = A058313((k+1)/2).

Original entry on oeis.org

1, 5, 7, 71, 379, 2659
Offset: 1

Views

Author

Alexander Adamchuk, Mar 10 2007

Keywords

Comments

For k > 1 all 5 listed terms are primes.
The only known numbers k such that k divides the numerator of alternating Harmonic number H'((k-1)/2) = A058313((k-1)/2) are the Wieferich primes (A001220): 1093 and 3511.
An odd prime p = prime(n) belongs to this sequence iff Fermat quotient A007663(n) == A130912(n) == 2*(-1)^((p+1)/2) (mod p). - Max Alekseyev, Nov 30 2022

Links

Crossrefs

Cf. A001008, A007663, A001220, A058313, A125854, A121999, A130912.

Programs

  • Mathematica
    f=0; Do[ f = f + (-1)^(n+1)*1/n; g = Numerator[f]; If[ IntegerQ[ g/(2n-1) ], Print[2n-1]], {n,1,3000} ]

Extensions

Edited by Max Alekseyev, Nov 30 2022
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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