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A182339 List of positive integers whose prime tower factorization, as defined in comments, contains the prime 2. 1
2, 4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 25, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 45, 46, 48, 49, 50, 52, 54, 56, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 88, 90, 92, 94, 96, 98, 99, 100, 102, 104, 106, 108, 110, 112 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

This set is the complement of A182318.

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:1204.5251 [math.NT], 2012-2014.

MAPLE

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

# conatinsPrimeInTower(2, n) returns true

conatinsPrimeInTower:=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:

MATHEMATICA

Select[Range[120], MemberQ[Flatten@ FixedPoint[Map[If[PrimeQ@ Last@# || Last@# == 1, #, {First@#, FactorInteger@Last@#}]&, #, {Depth@# - 2}]&, FactorInteger@#], 2]&] (* Jean-Fran├žois Alcover, Mar 27 2018, using Michael De Vlieger's program for A182318 )

CROSSREFS

Complement of A182318.

Sequence in context: A325572 A337945 A247423 * A110262 A176693 A118672

Adjacent sequences:  A182336 A182337 A182338 * A182340 A182341 A182342

KEYWORD

nonn

AUTHOR

Patrick Devlin, Apr 25 2012

STATUS

approved

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Last modified October 22 10:39 EDT 2021. Contains 348160 sequences. (Running on oeis4.)