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 A280617 Number of n-smooth Carmichael numbers. 1
 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 16, 16, 16, 16, 16, 16, 19, 19, 19, 19, 20, 20, 29, 29, 29, 29, 29, 29, 31, 31, 31, 31, 31, 31, 31, 31 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,17 LINKS Wikipedia, Carmichael number Wikipedia, Korselt's criterion Wikipedia, Smooth number EXAMPLE A number is b-smooth whenever its prime factors are all not greater than b. By Korselt's criterion, every Carmichael number is squarefree, therefore n-smooth numbers are products of subsets of primes below n. For n = 19, the n-smooth Carmichael numbers are 571 = 3*11*17 1105 = 5*13*17 1729 = 7*13*19 There are no other Carmichael numbers that are 19-smooth, therefore a(19)=3. MAPLE extend:= proc(c, p)      if andmap(q -> (p-1 mod q <> 0), c) then (c, c union {p}) else c fi end proc: carm:= proc(c) local n;   n:= convert(c, `*`);   map(t -> n-1 mod (t-1), c) = {0} end proc: A:= 0: A:= 0: Primes:= []: for k from 3 to 100 do   A[k]:= A[k-1];   if isprime(k) then     Cands:= {{}};     for i from 1 to nops(Primes) do       if (k - 1) mod Primes[i] <> 0 then         Cands:= map(extend, Cands, Primes[i])       fi;     od;     Cands:= map(`union`, select(c -> nops(c) > 1, Cands), {k});     A[k]:= A[k] + nops(select(carm, Cands));     Primes:= [op(Primes), k];   fi od: seq(A[i], i=1..100); # Robert Israel, Mar 14 2017 MATHEMATICA a[n_] := a[n] = If[n<17, 0, a[n-1] + If[! PrimeQ[n], 0, Block[{t, k = PrimePi@n, p}, p = Prime@k; Length@ Select[ Subsets[ Prime@ Range[2, k-1], {2, k-2}], (t = Times @@ #; Mod[t-1, p-1] == 0 && And @@ IntegerQ /@ ((p t - 1)/ (#-1))) &]]]]; Array[a, 80] (* Giovanni Resta, Mar 14 2017 *) PROG # (Python 3) from collections import Counter from functools import reduce from itertools import combinations, chain from operator import mul # http://www.sympy.org import sympy as sp limit = int(input()) # credit: http://stackoverflow.com/a/16915734 # as proved, there are no Carmichael numbers with # less than 3 prime factors, and thus modification def powerset(iterable):     xs = list(iterable)   return chain.from_iterable( combinations(xs, n) for n in range(3, len(xs)+1)) # all computed numbers will be limit-smooth def carmichael(limit):     for d in powerset(sp.primerange(3, limit)):       n = reduce(mul, d, 1)       broke = False       for p in d:           if (n-1)%(p-1) != 0:               broke = True               break       if not broke:             yield(d[-1]) # from list of pairs of (n, number_of_integers_with_n_as_greatest_prime_factor) # creates the sequence def prefix(lst):     r = []   s = 0   rpointer = 0   lstpointer = 0   while lstpointer < len(lst):       while lst[lstpointer] > rpointer:           r.append(s)           rpointer += 1       s += lst[lstpointer]       lstpointer += 1   r.append(s+lst[-1])   return r c = Counter(carmichael(limit)) for i, e in enumerate(prefix(sorted(c.items()))):     print(i, e) CROSSREFS Cf. A002997, A081702. Sequence in context: A214454 A140474 A091195 * A072375 A131981 A257244 Adjacent sequences:  A280614 A280615 A280616 * A280618 A280619 A280620 KEYWORD nonn AUTHOR Michal Radwanski, Jan 06 2017 STATUS approved

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Last modified May 26 18:08 EDT 2020. Contains 334630 sequences. (Running on oeis4.)