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A182337 List of positive integers whose prime tower factorization, as defined in comments, does not contain the prime 3.

%I A182337 #23 Apr 11 2020 06:10:23
%S A182337 1,2,4,5,7,10,11,13,14,16,17,19,20,22,23,25,26,28,29,31,32,34,35,37,
%T A182337 38,41,43,44,46,47,49,50,52,53,55,58,59,61,62,65,67,68,70,71,73,74,76,
%U A182337 77,79,80,82,83,85,86,89,91,92,94,95,97,98,100,101,103,106
%N A182337 List of positive integers whose prime tower factorization, as defined in comments, does not contain the prime 3.
%C A182337 The prime tower factorization of a number can be recursively defined as follows:
%C A182337 (0) The prime tower factorization of 1 is itself
%C A182337 (1) To find the prime tower factorization of an integer n>1, let n = p1^e1 * p2^e2 * ... * pk^ek be the usual prime factorization of n. Then the prime tower factorization is given by p1^(f1) * p2^(f2) * ... * pk^(fk), where fi is the prime tower factorization of ei.
%C A182337 As an alternative definition, let I(n) be the indicator function for the set of positive integers whose prime tower factorization does not contain a 3. Then I(n) is the multiplicative function satisfying I(p^k) = I(k) for p prime not equal to 3, and I(3^k) = 0.
%H A182337 Amiram Eldar, <a href="/A182337/b182337.txt">Table of n, a(n) for n = 1..10000</a>
%H A182337 Patrick Devlin and Edinah Gnang, <a href="https://arxiv.org/abs/1204.5251">Primes Appearing in Prime Tower Factorization</a>, arXiv:1204.5251 [math.NT], 2012-2014.
%p A182337 # The integer n is in this sequence if and only if
%p A182337 # containsPrimeInTower(3, n) returns false
%p A182337 containsPrimeInTower:=proc(q, n) local i, L, currentExponent; option remember;
%p A182337 if n <= 1 then return false: end if;
%p A182337 if type(n/q, integer) then return true: end if;
%p A182337 L := ifactors(n)[2];
%p A182337 for i to nops(L) do currentExponent := L[i][2];
%p A182337   if containsPrimeInTower(q, currentExponent) then return true: end if
%p A182337 end do;
%p A182337 return false:
%p A182337 end proc:
%p A182337 select(x-> not containsPrimeInTower(3,x), [$1..120])[];
%t A182337 indic[1] = 1; indic[n_] := indic[n] = Switch[f = FactorInteger[n], {{3, _}}, 0, {{_, _}}, indic[f[[1, 2]] ], _, Times @@ (indic /@ (Power @@@ f))]; Select[Range[120], indic[#] == 1&] (* _Jean-François Alcover_, Feb 25 2018 *)
%Y A182337 Cf. A182318.
%K A182337 nonn
%O A182337 1,2
%A A182337 _Patrick Devlin_, Apr 25 2012