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A182338 List of positive integers whose prime tower factorization, as defined in comments, contains the prime 3. 0
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..60.

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-Fran├žois Alcover, Jul 11 2018 *)

CROSSREFS

Complement of A182337.  Cf. A182318.

Sequence in context: A095277 A185717 A189637 * A267415 A140516 A310140

Adjacent sequences:  A182335 A182336 A182337 * A182339 A182340 A182341

KEYWORD

nonn

AUTHOR

Patrick Devlin, Apr 25 2012

STATUS

approved

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Last modified April 5 23:19 EDT 2020. Contains 333260 sequences. (Running on oeis4.)