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

1, 2, 4, 5, 7, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 37, 38, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 58, 59, 61, 62, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86, 89, 91, 92, 94, 95, 97, 98, 100, 101, 103, 106

Offset

1,2

Comments

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.

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.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Patrick Devlin and Edinah Gnang, Primes Appearing in Prime Tower Factorization, arXiv:1204.5251 [math.NT], 2012-2014.

Maple

# The integer n is in this sequence if and only if
# containsPrimeInTower(3, n) returns false
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-> not containsPrimeInTower(3, x), [$1..120])[];

Mathematica

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 *)

Crossrefs

Cf. A182318.

Sequence in context: A231009 A133254 A080725 * A024914 A189636 A117741

Adjacent sequences: A182334 A182335 A182336 * A182338 A182339 A182340

Keyword

nonn

Author

Patrick Devlin, Apr 25 2012

Status

approved