A182337 -id:A182337 - cp's OEIS frontend

A182338 List of positive integers whose prime tower factorization, as defined in comments, contains the prime 3.

3, 6, 8, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 40, 42, 45, 48, 51, 54, 56, 57, 60, 63, 64, 66, 69, 72, 75, 78, 81, 84, 87, 88, 90, 93, 96, 99, 102, 104, 105, 108, 111, 114, 117, 120, 123, 125, 126, 129, 132, 135, 136, 138, 141, 144, 147, 150, 152, 153

Offset: 1

Patrick Devlin, Apr 25 2012

nonn

This set is the complement of A182337.
The prime tower factorization of a number can be recursively defined as follows:
(0) The prime tower factorization of 1 is itself
(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.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Patrick Devlin and Edinah Gnang, Primes Appearing in Prime Tower Factorization, arXiv:1204.5251v1 [math.NT], 2012-2014.

Complement of A182337. Cf. A182318.

# The integer n is in this sequence if and only if
# containsPrimeInTower(3, n) returns true
containsPrimeInTower:=proc(q, n) local i, L, currentExponent; option remember;
if n <= 1 then return false: end if;
if type(n/q, integer) then return true: end if;
L := ifactors(n)[2];
for i to nops(L) do currentExponent := L[i][2];
if containsPrimeInTower(q, currentExponent) then return true: end if
end do;
return false:
end proc:
select(x-> containsPrimeInTower(3,x), [$1..160])[];

indic[1] = 1; indic[n_] := indic[n] = Switch[f = FactorInteger[n], {{3, }}, 0, {{, }}, indic[f[[1, 2]]], , Times @@ (indic /@ (Power @@@ f))];
Select[Range[200], indic[#] != 1&] (* Jean-François Alcover, Jul 11 2018 *)

cp's OEIS Frontend

A182338 List of positive integers whose prime tower factorization, as defined in comments, contains the prime 3.

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