A327787 a(n) is the smallest Carmichael number k such that gpf(p-1) = prime(n) for all prime factors p of k.

1729, 252601, 1152271, 1615681, 4335241, 172947529, 214852609, 79624621, 178837201, 775368901, 686059921, 985052881, 5781222721, 10277275681, 84350561, 5255104513, 492559141, 74340674101, 9293756581, 1200778753, 129971289169, 2230305949, 851703301, 8714965001, 6693621481

Offset

2,1

Comments

The first term is the Hardy-Ramanujan number. - Omar E. Pol, Nov 25 2019

Links

Amiram Eldar, Table of n, a(n) for n = 2..2905 (calculated using data from Claude Goutier; terms 2..831 from Daniel Suteu)

Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.

Daniel Suteu, Terms and upper bounds for n = 2..10000 (values greater than 2^64 are upper bounds).

Eric Weisstein's World of Mathematics, Carmichael Number.

Index entries for sequences related to Carmichael numbers.

Example

a(2) = 1729 = (2*3 + 1)(2*2*3 + 1)(2*3*3 + 1).
a(3) = 252601 = (2*2*2*5 + 1)(2*2*3*5 + 1)(2*2*5*5 + 1).
a(4) = 1152271 = (2*3*7 + 1)(2*3*3*7 + 1)(2*3*5*7 + 1).
a(5) = 1615681 = (2*11 + 1)(2*3*3*11 + 1)(2*2*2*2*2*11 + 1).

Mathematica

carmQ[n_] := CompositeQ[n] && Divisible[n - 1, CarmichaelLambda[n]]; gpf[n_] := FactorInteger[n][[-1, 1]]; g[n_] := If[Length[(u = Union[gpf /@ (FactorInteger[n][[;; , 1]] - 1)])] == 1, u[[1]], 1]; m = 5; c = 0; k = 0; v = Table[0, {m}]; While[c < m, k++ If[! carmQ[k], Continue[]]; If[(p = g[k]) > 1, i = PrimePi[p] - 1; If[i <= m && v[[i]] == 0, c++; v[[i]] = k]]]; v (* Amiram Eldar, Oct 08 2019 *)

Prog

(Perl) use ntheory ":all"; sub a { my $p = nth_prime(shift); for(my $k = 1; ; ++$k) { return $k if (is_carmichael($k) and vecall { (factor($_-1))[-1] == $p } factor($k)) } }
for my $n (2..10) { print "a($n) = ", a($n), "\n" }

Crossrefs

Cf. A002997 (Carmichael numbers), A006530 (gpf), A001235.

Sequence in context: A317126 A318646 A182087 * A352970 A033502 A277366

Adjacent sequences: A327784 A327785 A327786 * A327788 A327789 A327790

Keyword

nonn,changed

Author

Daniel Suteu, Sep 25 2019

Status

approved