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A283715 a(n) is the number of Carmichael numbers whose largest prime factor is prime(n).

%I A283715 #20 Sep 29 2024 21:10:04
%S A283715 0,0,0,0,0,0,2,1,0,1,2,2,2,0,0,0,0,6,3,1,9,2,0,3,9,7,3,1,16,20,42,19,
%T A283715 12,15,3,60,54,57,2,8,2,277,20,170,75,259,775,57,11,110
%N A283715 a(n) is the number of Carmichael numbers whose largest prime factor is prime(n).
%C A283715 Since Carmichael numbers are squarefree, there is only a finite number of them whose largest prime factor is any given prime.
%H A283715 Giovanni Resta, <a href="/A283715/a283715.txt">Carmichael numbers with largest prime factor <= 179, sorted by their largest prime factor</a>
%e A283715 a(28) = 1 because prime(28) = 107 and there is only one Carmichael number whose largest prime factor is 107, namely 413631505 = 5 * 7 * 17 * 73 * 89 * 107.
%t A283715 a[n_] := a[n] = If[n < 6, 0, Block[{t, p = Prime@ n}, Length@ Select[ Subsets[ Prime@ Range[2, n-1], {2, n-2}], (t = Times @@ #; Mod[t-1, p-1] == 0 && And @@ IntegerQ /@ ((p t - 1)/ (#-1))) &]]]; Array[a, 22]
%o A283715 (Python)
%o A283715 from math import prod
%o A283715 from itertools import combinations
%o A283715 from sympy import prime, primerange
%o A283715 def A283715(n):
%o A283715     plist, c = list(primerange(3,p:=prime(n))), 0
%o A283715     for l in range(2,len(plist)+1):
%o A283715         for q in combinations(plist,l):
%o A283715             k = prod(q)*p-1
%o A283715             if not (k%(p-1) or any(k%(r-1) for r in q)):
%o A283715                 c+=1
%o A283715     return c # _Chai Wah Wu_, Sep 25 2024
%Y A283715 Cf. A002997, A081702, A280617.
%K A283715 nonn,more
%O A283715 1,7
%A A283715 _Giovanni Resta_, Mar 15 2017
%E A283715 a(42)-a(50) from _Ondrej Kutal_, Sep 29 2024