A292367 Carmichael numbers c with record number of primes p such that c*p is also a Carmichael number.

561, 1729, 41041, 1615681, 14676481, 40622401, 173085121, 367804801, 631071001, 8494657921, 138399075361, 432081216001, 997803878401, 3837165696001, 7599525373441, 42182344790209, 65032633451521, 186137387251201, 329797704600001, 2523853463040001

Offset

1,1

Comments

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

The corresponding number of solutions is 0, 5, 7, 10, 12, 14, 18, 26, 30, 33, 55, 65, 71, 72, 90, 92, 112, 128, 192, 218.

Links

Table of n, a(n) for n=1..20.

Example

1729 has 5 prime numbers p: 37, 73, 109, 433 and 577, such that 1729*p: 63973, 126217, 188461, 748657 and 997633 are also 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={}; Do[n=A002997[[j]]; m=numSol[n]; If[m>numSolmax, AppendTo[seq, n]; numSolmax=m], {j, 1, Length[A002997]}]; seq

Crossrefs

Cf. A002997.

Sequence in context: A227976 A224930 A083732 * A290805 A217465 A097130

Adjacent sequences: A292364 A292365 A292366 * A292368 A292369 A292370

Keyword

nonn

Author

Amiram Eldar, Sep 15 2017

Status

approved