login
A182341
List of positive integers whose prime tower factorization, as defined in comments, contains the prime 7.
1
7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126, 128, 133, 140, 147, 154, 161, 168, 175, 182, 189, 196, 203, 210, 217, 224, 231, 238, 245, 252, 259, 266, 273, 280, 287, 294, 301, 308, 315, 322, 329, 336, 343, 350, 357, 364, 371, 378
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.
Observation: the union of 128 and the first 54 nonzero multiples of 7 (cf. A008589) gives the first 55 terms of this sequence. - Omar E. Pol, Feb 01 2020
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
# containsPrimeInTower(7, n) returns true
containsPrimeInTower:=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[400], containsPrimeInTower[7, #]&] (* Jean-François Alcover, Jan 22 2019, from Maple *)
CROSSREFS
Sequence in context: A044892 A004959 A020646 * A008589 A085130 A080194
KEYWORD
nonn
AUTHOR
Patrick Devlin, Apr 25 2012
STATUS
approved