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A283715
a(n) is the number of Carmichael numbers whose largest prime factor is prime(n).
3
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, 12, 15, 3, 60, 54, 57, 2, 8, 2, 277, 20, 170, 75, 259, 775, 57, 11, 110
OFFSET
1,7
COMMENTS
Since Carmichael numbers are squarefree, there is only a finite number of them whose largest prime factor is any given prime.
EXAMPLE
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.
MATHEMATICA
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]
PROG
(Python)
from math import prod
from itertools import combinations
from sympy import prime, primerange
def A283715(n):
plist, c = list(primerange(3, p:=prime(n))), 0
for l in range(2, len(plist)+1):
for q in combinations(plist, l):
k = prod(q)*p-1
if not (k%(p-1) or any(k%(r-1) for r in q)):
c+=1
return c # Chai Wah Wu, Sep 25 2024
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Giovanni Resta, Mar 15 2017
EXTENSIONS
a(42)-a(50) from Ondrej Kutal, Sep 29 2024
STATUS
approved