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.

A300748 Least prime divisor of A300629(n).

Original entry on oeis.org

3, 5, 7, 13, 17, 41, 43, 53, 101, 109, 113, 151, 211, 281, 307, 331, 337, 461, 617, 727, 739, 827, 1033, 1301, 1481, 1531, 1723, 1901, 2161, 2351, 2381, 2633, 2647, 2801, 3371, 3931, 4933, 5653, 5743, 6791, 6917, 7561, 8059, 9521, 10369, 11503, 11551, 12161, 17579, 17839, 18433, 20593, 21061, 23581, 26731, 30241
Offset: 1

Views

Author

Max Alekseyev, Mar 11 2018

Keywords

Comments

For m > 2, A135720(m) = A083876(m-1) if and only if a(n) = prime(m).

Crossrefs

Formula

a(n) = A020639(A300629(n)).

A083876 Least pseudoprime to base 2 through base prime(n).

Original entry on oeis.org

341, 1105, 1729, 29341, 29341, 162401, 252601, 252601, 252601, 252601, 252601, 252601, 1152271, 2508013, 2508013, 3828001, 3828001, 3828001, 3828001, 3828001, 3828001, 3828001, 3828001, 3828001, 3828001, 6733693, 6733693, 6733693
Offset: 1

Views

Author

Robert G. Wilson v, May 06 2003

Keywords

Comments

Records: 341, 1105, 1729, 29341, 162401, 252601, 1152271, 2508013, 3828001, 6733693, 17098369, 17236801, 29111881, 82929001, 172947529, 216821881, 228842209, 366652201, .... - Robert G. Wilson v, May 11 2012
Conjecture: for n > 1, a(n) is the smallest Carmichael number k with lpf(k) > prime(n). It seems that such Carmichael numbers have exactly three prime factors. - Thomas Ordowski, Apr 18 2017
The conjecture is true if a(n) < A285549(n) for all n > 1. It holds for all a(n) < 2^64. - Max Alekseyev and Thomas Ordowski, Mar 13 2018
If prime(n) < m < a(n), then m is prime if and only if p^(m-1) == 1 (mod m) for every prime p <= prime(n). - Thomas Ordowski, Mar 05 2018
By this conjecture in the second comment, a(n) <= A135720(n+1), with equality for n > 1 iff a(n) < a(n+1), namely for n = 2, 3, 5, 6, 12, 13, 15, 25, 28, 29, ... For such n, a(n) gives all terms of A300629 > 561. - Thomas Ordowski, Mar 10 2018

Crossrefs

Programs

  • Mathematica
    k = 4; Do[l = Table[ Prime[i], {i, 1, n}]; While[ PrimeQ[k] || Union[PowerMod[l, k - 1, k]] != {1}, k++ ]; Print[k], {n, 1, 29}]
  • PARI
    isps(k, n) = {if (isprime(k), return (0)); my(nbok = 0); for (b=2, prime(n), if (Mod(b, k)^(k-1) == 1, nbok++, break)); if (nbok==prime(n)-1, return (1));}
    a(n) = {my(k=2); while (!isps(k, n), k++); return (k);} \\ Michel Marcus, Apr 27 2018

A271221 Smallest Fermat pseudoprime k to all bases b = 2, 3, 4, ..., n.

Original entry on oeis.org

341, 1105, 1105, 1729, 1729, 29341, 29341, 29341, 29341, 29341, 29341, 162401, 162401, 162401, 162401, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601, 252601
Offset: 2

Views

Author

Felix Fröhlich, Apr 02 2016

Keywords

Comments

a(n) is the smallest composite k such that b^(k-1) == 1 (mod (b-1)k) for every b = 2, 3, 4, ..., n. For more comments, see A083876 and A300629. - Max Alekseyev and Thomas Ordowski, Apr 29 2018

Crossrefs

Programs

  • PARI
    a(n) = forcomposite(c=1, , my(i=0); for(b=2, n, if(Mod(b, c)^(c-1)==1, i++)); if(i==n-1, return(c)));

Extensions

Edited by Thomas Ordowski, Apr 29 2018
Corrected a typo within the initial terms by Jens Ahlström, Apr 23 2024

A348258 Records in A083876.

Original entry on oeis.org

341, 1105, 1729, 29341, 162401, 252601, 1152271, 2508013, 3828001, 6733693, 17098369, 17236801, 29111881, 82929001, 172947529, 216821881, 228842209, 366652201, 413138881, 2301745249, 2438403661, 5255104513, 5781222721, 8251854001, 12173703001, 13946829751, 15906120889, 23224518901
Offset: 1

Views

Author

Robert G. Wilson v, Oct 08 2021

Keywords

Crossrefs

Programs

  • Mathematica
    f[1] = 341; f[2] = 1105; f[3] = 1729; f[n_] := f[n] = Block[{k = f[n -1], lp = Prime@ Range[2, n -1], p = Prime@ n}, While[ PrimeQ@ k || PowerMod[2, k -1, k] != 1 || PowerMod[p, k -1, k] != 1 || Union[ PowerMod[lp, k -1, k]] != {1}, k += 2]; k]; Union[ Table[ f@n, {n, 1012}]]
Showing 1-4 of 4 results.