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.

A081702 Largest prime factor of the n-th Carmichael number (A002997).

Original entry on oeis.org

17, 17, 19, 29, 31, 41, 67, 73, 73, 61, 41, 97, 103, 89, 37, 31, 101, 241, 73, 233, 61, 109, 101, 113, 109, 397, 409, 67, 211, 137, 163, 181, 271, 421, 61, 197, 271, 199, 433, 73, 151, 61, 577, 271, 307, 37, 163, 211, 631, 541, 113, 353, 199, 331, 461, 101, 97
Offset: 1

Views

Author

Lekraj Beedassy, Apr 02 2003

Keywords

Links

Crossrefs

Cf. A002997, A006530.

Programs

  • Mathematica
    CarmichaelQ[n_] := Not[PrimeQ[n]] && Divisible[n - 1, CarmichaelLambda[n]]; FactorInteger[#][[-1, 1]]& /@ Select[Range[4, 10^7], CarmichaelQ] (* Amiram Eldar, Jun 24 2019 after Jean-François Alcover at A141710 *)

Formula

a(n) = A006530(A002997(n)). - Amiram Eldar, Jun 24 2019

A081703 Largest proper divisor of the n-th Carmichael number (A002997).

Original entry on oeis.org

187, 221, 247, 493, 403, 943, 1273, 2117, 2263, 2257, 5863, 3589, 7519, 20915, 9139, 6851, 14443, 8917, 18031, 9553, 24583, 26923, 6161, 55709, 7957, 24217, 17587, 26197, 12871, 10001, 89813, 13213, 16531, 40837, 42517, 218867, 59891, 95719
Offset: 1

Views

Author

Lekraj Beedassy, Apr 02 2003

Keywords

Examples

			The first Carmichael number is 3*11*17, so a(1) = 11*17.

Links

Crossrefs

Cf. A002997, A032742, A141710.

Programs

Formula

a(n) = A032742(A002997(n)) - David Wasserman, Jun 18 2004
a(n) = A002997(n)/A141710(n). - Robert Israel, Apr 13 2017

Extensions

More terms from David Wasserman, Jun 18 2004

A298701 Irregular triangle read by rows in which row n lists the prime factors of the n-th Carmichael number.

Original entry on oeis.org

3, 11, 17, 5, 13, 17, 7, 13, 19, 5, 17, 29, 7, 13, 31, 7, 23, 41, 7, 19, 67, 5, 29, 73, 7, 31, 73, 13, 37, 61, 7, 11, 13, 41, 13, 37, 97, 7, 73, 103, 3, 5, 47, 89, 7, 13, 19, 37, 11, 13, 17, 31, 7, 11, 13, 101, 13, 37, 241, 7, 13, 19, 73, 17, 41, 233, 7, 13, 31, 61
Offset: 1

Views

Author

Tim Johannes Ohrtmann, Jan 26 2018

Keywords

Comments

The n-th row is the A002997(n)-th row of A027746.
Length of the n-th row is A135717(n).
First term of the n-th row is A141710(n).
Last term of the n-th row is A081702(n).

Examples

			Array begins:
3, 11, 17,
5, 13, 17,
7, 13, 19,
5, 17, 29,
7, 13, 31.

Links

Crossrefs

Cf. A002997, A027746, A081702, A135717, A141710.

A321790 a(n) is the smallest base a > 2 such that a^(k-1) != 1 (mod k), where k = A001567(n), the n-th Fermat pseudoprime to base 2.

Original entry on oeis.org

3, 3, 3, 5, 3, 7, 3, 3, 5, 5, 7, 3, 3, 3, 3, 3, 3, 7, 3, 3, 3, 7, 3, 5, 3, 3, 3, 3, 3, 3, 3, 7, 3, 3, 5, 3, 3, 3, 3, 13, 3, 3, 3, 3, 5, 3, 3, 3, 3, 7, 3, 3, 13, 5, 3, 7, 3, 3, 3, 3, 3, 7, 3, 3, 3, 3, 3, 11, 3, 5, 5, 3, 3, 3, 5, 5, 3, 5, 7, 5, 5, 3, 13, 3, 3
Offset: 1

Views

Author

Thomas Ordowski, Nov 19 2018

Keywords

Comments

a(n) <= A177415(n).
Each a(n) is an odd prime.
If k = A001567(n) is a Carmichael number, then a(n) = lpf(k).
Conjecture: if k = A001567(n) is semiprime, then a(n) < lpf(k).
The smallest numbers k = A001567(n) such that a(n) = prime(m) for m > 1 are 341, 1105, 1729, 75361, 29341, 162401, 334153, ... See A135720 > 561.
The smallest such semiprimes are 341, 2701, ?, 721801, ... Cf. A285549.

Examples

			The first Fermat pseudoprime to base 2 is 341, and 341 is not a Fermat pseudoprime to base 3, so a(1) = 3.

Crossrefs

Cf. A001567, A002997, A083876, A135720, A177415, A214305, A285549.
A141710 is a subsequence.

Programs

  • Mathematica
    a[p_] := Module[{m=3}, While[Mod[m^(p-1), p] == 1, m++]; m]; psp = Select[Range[3, 1000000, 2], CompositeQ[ # ] && PowerMod[2, (# - 1), # ] == 1 &]; Map[a, psp] (* Amiram Eldar, Nov 19 2018 *)

Extensions

More terms from Amiram Eldar, Nov 19 2018
Showing 1-4 of 4 results.

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