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

%I A327787 #28 Feb 16 2025 08:33:58
%S A327787 1729,252601,1152271,1615681,4335241,172947529,214852609,79624621,
%T A327787 178837201,775368901,686059921,985052881,5781222721,10277275681,
%U A327787 84350561,5255104513,492559141,74340674101,9293756581,1200778753,129971289169,2230305949,851703301,8714965001,6693621481
%N A327787 a(n) is the smallest Carmichael number k such that gpf(p-1) = prime(n) for all prime factors p of k.
%C A327787 The first term is the Hardy-Ramanujan number. - _Omar E. Pol_, Nov 25 2019
%H A327787 Amiram Eldar, <a href="/A327787/b327787.txt">Table of n, a(n) for n = 2..2905</a> (calculated using data from Claude Goutier; terms 2..831 from Daniel Suteu)
%H A327787 Claude Goutier, <a href="http://www-labs.iro.umontreal.ca/~goutier/OEIS/A055553/">Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22</a>.
%H A327787 Daniel Suteu, <a href="/A327787/a327787.txt">Terms and upper bounds for n = 2..10000</a> (values greater than 2^64 are upper bounds).
%H A327787 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CarmichaelNumber.html">Carmichael Number</a>.
%H A327787 <a href="/index/Ca#Carmichael">Index entries for sequences related to Carmichael numbers</a>.
%e A327787 a(2) = 1729 = (2*3 + 1)(2*2*3 + 1)(2*3*3 + 1).
%e A327787 a(3) = 252601 = (2*2*2*5 + 1)(2*2*3*5 + 1)(2*2*5*5 + 1).
%e A327787 a(4) = 1152271 = (2*3*7 + 1)(2*3*3*7 + 1)(2*3*5*7 + 1).
%e A327787 a(5) = 1615681 = (2*11 + 1)(2*3*3*11 + 1)(2*2*2*2*2*11 + 1).
%t A327787 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 *)
%o A327787 (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)) } }
%o A327787 for my $n (2..10) { print "a($n) = ", a($n), "\n" }
%Y A327787 Cf. A002997 (Carmichael numbers), A006530 (gpf), A001235.
%K A327787 nonn
%O A327787 2,1
%A A327787 _Daniel Suteu_, Sep 25 2019