The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A182337 List of positive integers whose prime tower factorization, as defined in comments, does not contain the prime 3. 2
1, 2, 4, 5, 7, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 28, 29, 31, 32, 34, 35, 37, 38, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 58, 59, 61, 62, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 86, 89, 91, 92, 94, 95, 97, 98, 100, 101, 103, 106 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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.
As an alternative definition, let I(n) be the indicator function for the set of positive integers whose prime tower factorization does not contain a 3. Then I(n) is the multiplicative function satisfying I(p^k) = I(k) for p prime not equal to 3, and I(3^k) = 0.
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(3, n) returns false
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:
select(x-> not containsPrimeInTower(3, x), [$1..120])[];
MATHEMATICA
indic[1] = 1; indic[n_] := indic[n] = Switch[f = FactorInteger[n], {{3, _}}, 0, {{_, _}}, indic[f[[1, 2]] ], _, Times @@ (indic /@ (Power @@@ f))]; Select[Range[120], indic[#] == 1&] (* Jean-François Alcover, Feb 25 2018 *)
CROSSREFS
Cf. A182318.
Sequence in context: A231009 A133254 A080725 * A024914 A189636 A117741
KEYWORD
nonn
AUTHOR
Patrick Devlin, Apr 25 2012
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 14 02:53 EDT 2024. Contains 373392 sequences. (Running on oeis4.)