

A180044


Let the nth Carmichael number A002997(n) = p1*p2*...*pr, where p1 < p2 < ... < pr are primes. Then a(n) = (p11) * (p1*p2*...*pr  1)^(r2) / ((p21)*...*(pr1)).


0



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
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OFFSET

1,1


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 rfactor Carmichael number (p_1 < p_2 < ... < p_r). Then (p_11)*(k1)^(r2)/((p_21)*(p_31)*...*(p_r1)) is an integer. This was proved by Max Alekseyev (see link).
Contains A162290 as a subsequence.


LINKS

Table of n, a(n) for n=1..37.
Max Alekseyev, Pomerance's proof


EXAMPLE

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


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}](* JeanFrançois Alcover, Sep 28 2011 *)


PROG

(MAGMA) [ (d[1]1)*(n1)^(r2) / &*[ 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  (n1) mod (x1) eq 0} where r is #d where d is PrimeDivisors(n)]; // Klaus Brockhaus, Aug 10 2010


CROSSREFS

Cf. A104016, A104017, A002997, A162290.
Sequence in context: A101789 A174590 A162290 * A062725 A147121 A098334
Adjacent sequences: A180041 A180042 A180043 * A180045 A180046 A180047


KEYWORD

nonn


AUTHOR

A.K. Devaraj, Aug 08 2010


EXTENSIONS

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


STATUS

approved



