A182318 List of positive integers whose prime tower factorization, as defined in comments, does not contain the prime 2.

1, 3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 119, 123, 125, 127, 129, 131, 133, 135, 137, 139, 141, 143, 145, 149

Offset: 1

Author

Patrick Devlin, Apr 24 2012

Keywords

• nonn

Comments

The prime tower factorization of a number can be recursively defined as follows: the prime tower factorization of 1 is itself; to find the prime tower factorization of an integer n > 1, let n = p_1^e_1 * p_2^e_2 * ... * p_k^e_k be the canonical prime factorization of n, then the prime tower factorization is given by p_1^f_1 * p_2^f_2 * ... * p_k^f_k, where f_i is the prime tower factorization of e_i.

An alternative definition: let I(n) be the indicator function for the set of positive integers whose prime tower factorization does not contain a 2. Then I(n) is the multiplicative function satisfying I(p^k) = I(k) for p prime not equal to 2, and I(2^k) = 0.

Links

• Rémy Sigrist, 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.

Crossrefs

A276378 is a subsequence.

Programs

Maple

# The integer n is in this sequence if and only if
# containsPrimeInTower(2, 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:

Mathematica

Select[Range[150], ! MemberQ[Flatten@ FixedPoint[Map[If[PrimeQ@ Last@ # || Last@ # == 1, #, {First@ #, FactorInteger@ Last@ #}] &, #, {Depth@ # - 2}] &, FactorInteger@ #], 2] &] (* Michael De Vlieger, Feb 17 2017 *)
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[150], !containsPrimeInTower[2, #]&] (* Jean-François Alcover, Jan 22 2019, translated from Maple *)

PARI

is(n)=if(n<4, return(n!=2)); if(n%2==0, return(0)); my(f=factor(n)[,2]); for(i=1,#f, if(!is(f[i]), return(0))); 1 \\ Charles R Greathouse IV, May 16 2024

Extensions

Typo in Maple program corrected by Rémy Sigrist, Dec 13 2016