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.

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

Links

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

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

Cf. A052155, A083737, A083739, A083876, A300629.

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

A230746 Carmichael numbers of the form (30*k + 1)*(120*k + 1)*(150*k + 1), where 30*k + 1, 120*k + 1 and 150*k + 1 are all primes.

Original entry on oeis.org

68154001, 3713287801, 63593140801, 122666876401, 193403531401, 227959335001, 246682590001, 910355497801, 4790779641001, 5367929037001, 6486222838801, 24572944746001, 25408177226401, 27134994772801, 55003376283001, 63926508701401, 108117809748001, 112614220996801
Offset: 1

Views

Author

Arkadiusz Wesolowski, Oct 29 2013

Keywords

Links

Crossrefs

Subsequence of A083739 and of A230722.
Cf. A002997, A007304, A157956.

Programs

  • Magma
    [n : k in [1..593 by 2] | IsPrime(a) and IsPrime(b) and IsPrime(c) and IsOne(n mod CarmichaelLambda(n)) where n is a*b*c where a is 30*k+1 where b is 120*k+1 where c is 150*k+1]
  • Mathematica
    carmQ[n_] := CompositeQ[n] && Divisible[n - 1, CarmichaelLambda[n]]; v = {30, 120, 150}; Times @@ (v*# + 1) & /@ Select[Range[1000], AllTrue[(w = v*# + 1), PrimeQ] && carmQ[Times @@ w] &] (* Amiram Eldar, Nov 11 2019 *)

Formula

(A007304 INTERSECT A157956) INTERSECT A230722.

A114250 Number of Fermat pseudoprimes to bases 2, 3, 5 and 7 less than 10^n.

Original entry on oeis.org

0, 0, 0, 0, 3, 19, 63, 175, 501, 1230, 3086, 7469, 18402, 44748, 109787, 269289, 668521, 1675317, 4236270
Offset: 1

Views

Author

Eric W. Weisstein, Nov 18 2005

Keywords

Links

Crossrefs

Cf. A083739.

Programs

  • Mathematica
    Table[Count[Select[Range[2, 10^6], ! PrimeQ[#] && PowerMod[2, # - 1, #] == 1 && PowerMod[3, # - 1, #] == 1 && PowerMod[5, # - 1, #] == 1 && PowerMod[7, # - 1, #] == 1 &], x_ /; x < 10^n], {n, 6}]  (* Robert Price, Jun 09 2019 *)

Formula

a(n) = card{ m in A083739, m<10^n}. - R. J. Mathar, Feb 07 2008

Extensions

a(9)-a(12) from Amiram Eldar, Sep 18 2021
a(13)-a(19) from Amiram Eldar, Apr 22 2022
Showing 1-4 of 4 results.

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