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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A334048 Primes p that set a new record for the number of bases 1 < b < p for which p is a base-b Wieferich prime and b is not a perfect power.

Original entry on oeis.org

2, 11, 71, 269, 653, 5107, 103291, 728471, 2544079
Offset: 1

Views

Author

Jeppe Stig Nielsen, Sep 06 2020

Keywords

Comments

p is a base-b Wieferich prime iff b^(p-1) == 1 (mod p^2).
Records in A248865 sometimes arise when all the b values (bases) are powers of the same small integer. By excluding powers, we find primes that are Wieferich in many "independent" ways.

Examples

			Prime 5107 is Wieferich to six bases (under 5107), namely 560, 1209, 1779, 2621, 4295, 4361, none of which are perfect powers. A prime such as 1093 is Wieferich to ten bases, namely 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024; however, when dismissing perfect powers, only one of the ten bases is left. In fact, no prime less than 5107 has six or more bases when perfect powers are dismissed, so 5107 sets a record and is included in this sequence.

Crossrefs

Cf. A134307, A175932, A242830, A248865.

Programs

  • PARI
    r=-oo; forprime(p=2,, i=sum(b=2,p-1,!ispower(b) && Mod(b,p^2)^(p-1)==1); if(i>r, print1(p, ", "); r=i))

Extensions

a(8)-a(9) from Kellen Shenton added by Jeppe Stig Nielsen, Sep 12 2020

A259505 Primes p where an integer r with 1 < r < p exists such that r^r == 1 (mod p^2).

Original entry on oeis.org

997, 8647, 47521, 85991, 283411, 4594451, 5476381, 52728733, 53920829, 100980223, 127072849, 436118521, 585293099, 858905011
Offset: 1

Views

Author

Felix Fröhlich, Nov 08 2015

Keywords

Comments

Inspired by the Nov 28 2014 comment from Thomas Ordowski in A001220.
a(5) > 276929 if it exists.
a(15) > 10^9. - Hiroaki Yamanouchi, Dec 19 2015

Examples

			252^252 = 1 mod 997^2.
5764^5764 = 1 mod 8647^2.

Crossrefs

Cf. A001220, A134307.

Programs

  • Mathematica
    p = 2; lst = {}; While[p < 100001, r = 2; While[r < p, If[ PowerMod[r, r, p^2] == 1, AppendTo[lst, p]]; r++]; p = NextPrime@ p] (* Robert G. Wilson v, Dec 06 2015 *)
  • PARI
    forprime(p=1, , for(r=2, p-1, if(Mod(r, p^2)^r==1, print1(p, ", "); break({1}))))

Extensions

a(5)-a(14) from Hiroaki Yamanouchi, Dec 19 2015

A275338 Smallest prime p where a base b with 1 < b < p exists such that b^(p-1) == 1 (mod p^n).

Original entry on oeis.org

3, 11, 113
Offset: 1

Views

Author

Felix Fröhlich, Jul 28 2016

Keywords

Comments

Smallest prime p such that A254444(i) >= n, where i is the index of p in A000040.
For n > 1, a(n) is a term of A134307.
For n > 1, if A000040(i) is a term of the sequence, then A249275(i) < A000040(i).
For n > 1, smallest prime p such that T(n, i) < p, where i is the index of p in A000040 and T = A257833.
a(4) > 5*10^8 if it exists (see Fischer link).

Examples

			For n = 3: p = 113 satisfies 68^(p-1) == 1 (mod p^3) and there is no smaller prime p such that p satisfies b^(p-1) == 1 (mod p^3) for some b with 1 < b < p, so a(3) = 113.

Links

Crossrefs

Cf. A134307, A254444, A257833.

Programs

  • PARI
    a(n) = forprime(p=1, , for(b=2, p-1, if(Mod(b, p^n)^(p-1)==1, return(p))))
Previous Showing 11-13 of 13 results.

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

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