A182341 List of positive integers whose prime tower factorization, as defined in comments, contains the prime 7.

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

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
# 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

Cf. A008589, A182318.

Sequence in context: A044892 A004959 A020646 * A008589 A085130 A080194

Adjacent sequences: A182338 A182339 A182340 * A182342 A182343 A182344

Keyword

nonn

Author

Patrick Devlin, Apr 25 2012

Status

approved