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A182340
List of positive integers whose prime tower factorization, as defined in comments, contains the prime 5.
1
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
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
KEYWORD
nonn
AUTHOR
Patrick Devlin, Apr 25 2012
STATUS
approved