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

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

Author

Robert G. Wilson v, May 06 2003

Keywords

• nonn

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

Links

• Giovanni Resta, Table of n, a(n) for n = 1..1000 (first 100 terms from Robert G. Wilson v)

Crossrefs

Cf. A002997, A001567, A052155, A083737, A087788, A083739, A135720, A141768, A250199, A271221, A285549, A300629.

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