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

1,1

Comments

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.

Links

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.

Maple

# 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])[];

Mathematica

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 *)

Crossrefs

Complement of A182337. Cf. A182318.

Sequence in context: A095277 A185717 A189637 * A267415 A376279 A140516

Adjacent sequences: A182335 A182336 A182337 * A182339 A182340 A182341

Keyword

nonn

Author

Patrick Devlin, Apr 25 2012

Status

approved