A182339 - OEIS

%I #15 Apr 11 2020 06:10:37

%S 2,4,6,8,9,10,12,14,16,18,20,22,24,25,26,28,30,32,34,36,38,40,42,44,

%T 45,46,48,49,50,52,54,56,58,60,62,63,64,66,68,70,72,74,75,76,78,80,81,

%U 82,84,86,88,90,92,94,96,98,99,100,102,104,106,108,110,112

%N List of positive integers whose prime tower factorization, as defined in comments, contains the prime 2.

%C This set is the complement of A182318.

%C The prime tower factorization of a number can be recursively defined as follows:

%C (0) The prime tower factorization of 1 is itself

%C (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.

%H Amiram Eldar, <a href="/A182339/b182339.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 # The integer n is in this sequence if and only if

%p # conatinsPrimeInTower(2, n) returns true

%p conatinsPrimeInTower:=proc(q, n) local i, L, currentExponent; option remember;

%p if n <= 1 then return false: end if;

%p if type(n/q, integer) then return true: end if;

%p L := ifactors(n)[2];

%p for i to nops(L) do currentExponent := L[i][2];

%p if containsPrimeInTower(q, currentExponent) then return true: end if

%p end do;

%p return false:

%p end proc:

%t Select[Range[120], MemberQ[Flatten@ FixedPoint[Map[If[PrimeQ@ Last@# || Last@# == 1, #, {First@#, FactorInteger@Last@#}]&, #, {Depth@# - 2}]&, FactorInteger@#], 2]&] (* _Jean-François Alcover_, Mar 27 2018, using _Michael De Vlieger_'s program for A182318 )

%Y Complement of A182318.

%K nonn

%O 1,1

%A _Patrick Devlin_, Apr 25 2012