A182340 List of positive integers whose prime tower factorization, as defined in comments, contains the prime 5.

5, 10, 15, 20, 25, 30, 32, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 96, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 220, 224, 225, 230, 235, 240, 243, 245, 250, 255, 260

Offset

1,1

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.

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(5, 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

containsPrimeInTower[q_, n_] := containsPrimeInTower[q, n] = Module[{i, L, currentExponent}, If[n <= 1, Return[False]]; If[IntegerQ[n/q], Return[True]]; L = FactorInteger[n]; For[i = 1, i <= Length[L], i++, currentExponent = L[[i, 2]]; If[containsPrimeInTower[q, currentExponent], Return[True]]]; Return[False]];
Select[Range[300], containsPrimeInTower[5, #]&] (* Jean-François Alcover, Jan 22 2019, from Maple *)

Crossrefs

Cf. A182318.

Sequence in context: A313732 A306236 A297305 * A313733 A076311 A336483

Adjacent sequences: A182337 A182338 A182339 * A182341 A182342 A182343

Keyword

nonn

Author

Patrick Devlin, Apr 25 2012

Status

approved