A180044 Let the n-th Carmichael number A002997(n) = p1*p2*...*pr, where p1 < p2 < ... < pr are primes. Then a(n) = (p1-1) * (p1*p2*...*pr - 1)^(r-2) / ((p2-1)*...*(pr-1)).

7, 23, 48, 22, 47, 45, 45, 21, 44, 163, 2105352, 162, 43, 486266, 3157729, 9859600, 5110605, 161, 6146018, 280, 8225424, 9135075, 1684, 6185169, 1363, 159, 351, 59907600, 950, 1675, 9879408, 1358, 949, 158, 95468562, 4399220, 83722500

Offset: 1

Author

A.K. Devaraj, Aug 08 2010

Keywords

• nonn

Comments

a(n) is always an integer as proved at the Alekseyev link.

The conjecture referred to in A162290 was generalized as follows: Let k be an r-factor Carmichael number (p_1 < p_2 < ... < p_r). Then (p_1-1)*(k-1)^(r-2)/((p_2-1)*(p_3-1)*...*(p_r-1)) is an integer. This was proved by Max Alekseyev (see link).

Contains A162290 as a subsequence.

Examples

			Since A002997(11) = 41041 = 7*11*13*41, we have a(11) = (6*41040^2) / (10*12*40) = 2105352.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Max Alekseyev, Pomerance's proof.

Crossrefs

Cf. A104016, A104017, A002997, A162290.

Programs

Magma

[ (d[1]-1)*(n-1)^(r-2) / &*[ d[i]-1: i in [2..r] ]: n in [3..700000 by 2] | not IsPrime(n) and IsSquarefree(n) and forall(t){x: x in d | (n-1) mod (x-1) eq 0} where r is #d where d is PrimeDivisors(n)]; // Klaus Brockhaus, Aug 10 2010

Mathematica

lim = 1000001; CarmichaelQ[n_] := Divisible[n - 1, CarmichaelLambda[n]] && ! PrimeQ[n]; cc = Select[Table[k, {k, 561, lim, 2}], CarmichaelQ]; lg = Length[cc]; a[n_] := (c = cc[[n]]; pp = FactorInteger[c][[All, 1]]; r = Length[pp]; (pp[[1]] - 1)*((Times @@ pp - 1)^(r - 2)/ Times @@ (Drop[pp, 1] - 1))); Table[a[n], {n, 1, lg}] (* Jean-François Alcover, Sep 28 2011 *)

Extensions

Edited and extended by Max Alekseyev and Klaus Brockhaus, Aug 10 2010