A292366 Numbers n with record number of primes p such that n*p is a Carmichael number.

1, 33, 85, 481, 1729, 20801, 1615681, 14676481, 40622401, 93980251, 367804801, 631071001, 8494657921, 138399075361

Offset

1,2

Comments

If n*p is a Carmichael number, where p is a prime, then (p-1)|(n-1), so given n, the number of possible primes is bounded by the number of divisors of n-1.

The corresponding number of solutions is 0, 1, 2, 3, 5, 7, 10, 12, 14, 18, 26, 30, 33, 55.

Links

Table of n, a(n) for n=1..14.

Example

33 has 1 prime number, 17, such that 33*17 = 561 is a Carmichael number.
85 has 2 prime numbers, 13 and 29, such that 85*13 = 1105 and 85*29 = 2465 are Carmichael numbers.

Mathematica

carmichaelQ[n_]:= Not[PrimeQ[n]] && Divisible[n-1, CarmichaelLambda[n]];
numSol[n_] := Module[{m = 0}, ds = Divisors[n-1]; Do[p = ds[[k]] + 1; If[! PrimeQ[p], Continue[]]; If[! carmichaelQ[p*n], Continue[]]; m++, {k, 1, Length[ds] - 1}]; m]; numSolmax = -1; seq = {}; nums = {}; Do[m = numSol[n]; If[m > numSolmax, AppendTo[seq, n]; AppendTo[nums, m]; numSolmax = m], {n, 1, 10^8}]; seq

Crossrefs

Cf. A002997.

Sequence in context: A355711 A250732 A080700 * A080200 A067705 A075213

Adjacent sequences: A292363 A292364 A292365 * A292367 A292368 A292369

Keyword

nonn,more

Author

Amiram Eldar, Sep 15 2017

Extensions

a(11)-a(14) from Giovanni Resta, May 08 2020

Status

approved