A182339 List of positive integers whose prime tower factorization, as defined in comments, contains the prime 2.

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

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-François Alcover, Mar 27 2018, using Michael De Vlieger's program for A182318 )

Crossrefs

Complement of A182318.

Sequence in context: A337945 A372771 A247423 * A110262 A176693 A352872

Adjacent sequences: A182336 A182337 A182338 * A182340 A182341 A182342

Keyword

nonn

Author

Patrick Devlin, Apr 25 2012

Status

approved