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 1-10 of 12 results. Next

A255920 Number of primes p with p < n such that n^(p-1) == 1 (mod p^2) i.e., number of Wieferich primes to base n less than n.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 2, 2, 3, 0, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 0, 1, 0, 2, 1, 0, 2, 2, 1, 1, 1, 1, 1, 0, 1, 2, 1, 2, 0, 3, 1, 1, 0, 2, 0, 0, 1, 1, 2, 2, 1, 2, 0, 2, 3, 2, 1, 2, 0, 2, 1, 2, 2, 1, 1, 1, 3, 2, 2, 0, 0, 1, 0, 0, 0
Offset: 2

Views

Author

Felix Fröhlich, Mar 11 2015

Keywords

Links

Crossrefs

Cf. A242830.

Programs

  • Mathematica
    f[n_] := Block[{p = Complement[Prime@ Range@ PrimePi@ n, First /@ FactorInteger@ n]}, Select[p, Divisible[n^(# - 1) - 1, #^2] &]]; Length /@ Table[f@ n, {n, 2, 120}] (* Michael De Vlieger, Sep 24 2015 *)
  • PARI
    for(n=2, 120, i=0; forprime(p=1, n, if(Mod(n, p^2)^(p-1)==1, i++)); print1(i, ", "))

A255203 Primes p for which no bases b with 1 < b < p exist such that p is a base b Wieferich prime.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 23, 31, 41, 47, 53, 61, 67, 73, 83, 89, 101, 107, 139, 149, 157, 167, 173, 179, 193, 227, 239, 251, 271, 277, 311, 317, 337, 383, 389, 409, 431, 443, 457, 467, 479, 491, 503, 541, 569, 587, 593, 613, 643, 677, 683, 691, 709, 719, 733
Offset: 1

Views

Author

Felix Fröhlich, Feb 17 2015

Keywords

Comments

p = prime(n) such that A242830(n) = 0.

Crossrefs

Cf. A255204, A255205, A255206, A255207, A255208, A255209, A255210.

Programs

  • PARI
    forprime(p=1, , b=2; i=0; while(b < p, if(Mod(b, p^2)^(p-1)==1, i++); b++); if(i==0, print1(p, ", ")))

A255204 Primes p for which exactly one base b with 1 < b < p exists such that p is a base b Wieferich prime.

Original entry on oeis.org

29, 37, 43, 59, 79, 97, 103, 109, 113, 137, 151, 181, 191, 197, 199, 223, 233, 241, 257, 263, 281, 283, 293, 307, 373, 379, 397, 401, 419, 421, 433, 439, 449, 461, 499, 521, 523, 547, 557, 563, 577, 601, 617, 619, 641, 659, 661, 701, 727, 739, 743, 761, 769
Offset: 1

Views

Author

Felix Fröhlich, Feb 17 2015

Keywords

Comments

p = prime(n) such that A242830(n) = 1.

Crossrefs

Cf. A255203, A255205, A255206, A255207, A255208, A255209, A255210.

Programs

  • PARI
    forprime(p=1, , b=2; i=0; while(b < p, if(Mod(b, p^2)^(p-1)==1, i++); b++); if(i==1, print1(p, ", ")))

A255205 Primes p for which exactly two bases b with 1 < b < p exist such that p is a base b Wieferich prime.

Original entry on oeis.org

11, 71, 127, 131, 163, 211, 229, 313, 347, 349, 353, 359, 367, 463, 509, 599, 607, 631, 647, 673, 797, 827, 829, 977, 1021, 1061, 1087, 1109, 1123, 1187, 1327, 1381, 1399, 1429, 1453, 1483, 1493, 1499, 1523, 1531, 1549, 1553, 1607, 1613, 1619, 1621, 1657, 1669
Offset: 1

Views

Author

Felix Fröhlich, Feb 17 2015

Keywords

Comments

p = prime(n) such that A242830(n) = 2.

Crossrefs

Cf. A255203, A255204, A255206, A255207, A255208, A255209, A255210.

Programs

  • PARI
    forprime(p=1, , b=2; i=0; while(b < p, if(Mod(b, p^2)^(p-1)==1, i++); b++); if(i==2, print1(p, ", ")))

A255206 Primes p for which exactly three bases b with 1 < b < p exist such that p is a base b Wieferich prime.

Original entry on oeis.org

269, 331, 571, 863, 883, 907, 1097, 1103, 1291, 1579, 1697, 1741, 2179, 2213, 2221, 2281, 2309, 2311, 2551, 2671, 2677, 2693, 2707, 2789, 2791, 3191, 3253, 3571, 3617, 3877, 3931, 4049, 4787, 4813, 4987, 5021, 5153, 5197, 5227, 5347, 5519, 5669, 5689, 5693
Offset: 1

Views

Author

Felix Fröhlich, Feb 17 2015

Keywords

Comments

p = prime(n) such that A242830(n) = 3.

Crossrefs

Cf. A255203, A255204, A255205, A255207, A255208, A255209, A255210.

Programs

  • PARI
    forprime(p=1, , b=2; i=0; while(b < p, if(Mod(b, p^2)^(p-1)==1, i++); b++); if(i==3, print1(p, ", ")))

A255207 Primes p for which exactly four bases b with 1 < b < p exist such that p is a base b Wieferich prime.

Original entry on oeis.org

487, 1163, 2203, 2731, 3373, 4391, 5261, 6343, 6451, 6569, 7753, 8377, 8863, 9041, 9397, 9463, 9941, 10079, 10321, 11783, 12703, 13121, 13151, 13807, 13903, 14419, 15061, 15263, 15313, 15601, 16631, 16883, 16943, 17477, 17519, 18253, 18773, 20173, 22279, 23291
Offset: 1

Views

Author

Felix Fröhlich, Feb 17 2015

Keywords

Comments

p = prime(n) such that A242830(n) = 4.

Crossrefs

Cf. A255203, A255204, A255205, A255206, A255208, A255209, A255210.

Programs

  • PARI
    forprime(p=1, , b=2; i=0; while(b < p, if(Mod(b, p^2)^(p-1)==1, i++); b++); if(i==4, print1(p, ", ")))

A255208 Primes p for which exactly five bases b with 1 < b < p exist such that p is a base b Wieferich prime.

Original entry on oeis.org

653, 4909, 7723, 9811, 13691, 15413, 18133, 18223, 21061, 22147, 25679, 29131, 33923, 35353, 36913, 37633, 46021, 57527, 61819, 66107, 71059, 72643, 77867, 79867, 85061, 87509, 89113, 96757, 97213, 98641, 117977, 118163, 120247, 122209, 123653, 126443, 129061
Offset: 1

Views

Author

Felix Fröhlich, Feb 17 2015

Keywords

Comments

p = prime(n) such that A242830(n) = 5.

Crossrefs

Cf. A255203, A255204, A255205, A255206, A255207, A255209, A255210.

Programs

  • PARI
    forprime(p=1, , b=2; i=0; while(b < p, if(Mod(b, p^2)^(p-1)==1, i++); b++); if(i==5, print1(p, ", ")))

A255209 Primes p for which exactly six bases b with 1 < b < p exist such that p is a base b Wieferich prime.

Original entry on oeis.org

5107, 20771, 51427, 52517, 61417, 66161, 116731, 119359, 128657, 140741, 147647, 150559, 199783, 203773, 213949, 229939, 237283, 261761, 286751, 288929, 303089, 339139, 342373, 381853, 384611, 385657, 475897
Offset: 1

Views

Author

Felix Fröhlich, Feb 17 2015

Keywords

Comments

p = prime(n) such that A242830(n) = 6.

Links

Crossrefs

Cf. A255203, A255204, A255205, A255206, A255207, A255208, A255210.

Programs

  • PARI
    forprime(p=1, , b=2; i=0; while(b < p, if(Mod(b, p^2)^(p-1)==1, i++); b++); if(i==6, print1(p, ", ")))

A255210 Primes p for which exactly seven bases b with 1 < b < p exist such that p is a base-b Wieferich prime.

Original entry on oeis.org

103291, 491531, 534851, 804367, 997961, 1026899, 1062427, 1457389, 1550513, 2327629, 2602307, 3093121, 3137257, 3181481, 3412741, 3497381, 3720179, 3814253, 4087301, 4234057, 4891973, 5063087, 5131237, 5194789, 5736611, 6253349, 6903191, 6906469, 6945047
Offset: 1

Views

Author

Felix Fröhlich, Mar 07 2015

Keywords

Comments

p = prime(n) such that A242830(n) = 7.

Links

Crossrefs

Cf. A255203, A255204, A255205, A255206, A255207, A255208, A255209.

Programs

  • PARI
    forprime(p=1, , b=2; i=0; while(b < p, if(Mod(b, p^2)^(p-1)==1, i++); b++); if(i==7, print1(p, ", ")))

A248865 Primes p that set a new record for the number of bases 1 < b < p for which p is a base-b Wieferich prime.

Original entry on oeis.org

2, 11, 269, 487, 653, 1093, 3511, 1006003
Offset: 1

Views

Author

Felix Fröhlich, Mar 07 2015

Keywords

Comments

Primes p where A242830(i) reaches record values, where i is the index of p in A000040.
The corresponding number of bases are 0, 2, 3, 4, 5, 10, 11, 12. - Jianing Song, Feb 07 2019
From Jeppe Stig Nielsen, Sep 06 2020: (Start)
Note that for a(6) and a(7), all the b values (bases) that are counted are powers of 2; and for a(8) all are powers of 3.
See A334048 for a version where bases b that are powers are not allowed.
One candidate for a(9) is 1645333507; it has 14 bases, the first 13 of which are powers of 5. However, excluding all numbers under 1645333507 as candidates for a(9) may be difficult to do.
(End)

Crossrefs

Cf. A000040, A175932, A242830, A334048.
Subsequence of A175932. Apart from the first term, subsequence of A134307.

Programs

  • PARI
    my(r=-1); forprime(p=2, , my(b=2, i=0); while(b < p, if(Mod(b, p^2)^(p-1)==1, i++); b++); if(i > r, print1(p, ", "); r=i)) \\ changed to include a(1) = 2 by Jianing Song, Feb 07 2019

Extensions

a(1) = 2 inserted by Jianing Song, Feb 07 2019
Showing 1-10 of 12 results. Next

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

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