A182341 - OEIS

%I #28 Apr 11 2020 06:40:14

%S 7,14,21,28,35,42,49,56,63,70,77,84,91,98,105,112,119,126,128,133,140,

%T 147,154,161,168,175,182,189,196,203,210,217,224,231,238,245,252,259,

%U 266,273,280,287,294,301,308,315,322,329,336,343,350,357,364,371,378

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

%C The prime tower factorization of a number can be recursively defined as follows:

%C (0) The prime tower factorization of 1 is itself

%C (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.

%C 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

%H Amiram Eldar, <a href="/A182341/b182341.txt">Table of n, a(n) for n = 1..10000</a>

%H Patrick Devlin and Edinah Gnang, <a href="http://arxiv.org/abs/1204.5251">Primes Appearing in Prime Tower Factorization</a>, arXiv:1204.5251 [math.NT], 2012-2014.

%p # The integer n is in this sequence if and only if

%p # containsPrimeInTower(7, n) returns true

%p containsPrimeInTower:=proc(q, n) local i, L, currentExponent; option remember;

%p if n <= 1 then return false: end if;

%p if type(n/q, integer) then return true: end if;

%p L := ifactors(n)[2];

%p for i to nops(L) do currentExponent := L[i][2];

%p if containsPrimeInTower(q, currentExponent) then return true: end if

%p end do;

%p return false:

%p end proc:

%t 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]];

%t Select[Range[400], containsPrimeInTower[7, #]&] (* _Jean-François Alcover_, Jan 22 2019, from Maple *)

%Y Cf. A008589, A182318.

%K nonn

%O 1,1

%A _Patrick Devlin_, Apr 25 2012