A080259 Numbers whose squarefree kernel is not a primorial number, i.e., A007947(a(n)) is not in A002110.

3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87

Offset: 1

Author

Labos Elemer, Mar 19 2003

Keywords

• nonn

Comments

Complement to A055932.

From Michael De Vlieger, Feb 06 2024: (Start)

Odd prime power p^m, m >= 1 is in the sequence since its squarefree kernel p is odd and not a primorial. Therefore 3^3, 5^2, etc. are in the sequence.

Odd squarefree composite k is in the sequence since its squarefree kernel is odd and thus not a primorial. Therefore 15 and 33 are in the sequence.

Numbers k such that A053669(k) < A006530(k) are in the sequence since the condition A053669(k) < A006530(k) implies the squarefree kernel is not a primorial, etc. (End)

Examples

			From _Michael De Vlieger_, Jan 23 2024: (Start)
1 is not in the sequence because its squarefree kernel is 1, the product of the 0 primes that divide 1 (the "empty product") and therefore the same as A002110(0), the 0th primorial.
2 is not in the sequence since its squarefree kernel is 2, the smallest prime, hence the same as A002110(1) = 2.
4 is not in the sequence since its squarefree kernel is 2 = A002110(1).
(End)
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002110, A006530, A007947, A053669, A055932.

Programs

Mathematica

Select[Range[120], Nor[IntegerQ@ Log2[#], And[EvenQ[#], Union@ Differences@ PrimePi[FactorInteger[#][[All, 1]]] == {1}]] &] (* Michael De Vlieger, Jan 23 2024 *)

PARI

is(n) = {my(f=factor(n)[,1]);n>1&&primepi(f[#f])>#f} \\ David A. Corneth, May 22 2016

Python

from itertools import count, islice
from sympy import primepi, primefactors
def A080259_gen(startvalue=2): # generator of terms >= startvalue
    for k in count(max(startvalue,2)):
        p = list(map(primepi,primefactors(k)))
        if not(min(p)==1 and max(p)==len(p)):
            yield k
A080259_list = list(islice(A080259_gen(),40)) # Chai Wah Wu, Aug 07 2025

Formula

{a(n)} = { k : A053669(k) < A006530(k) }. - Michael De Vlieger, Jan 23 2024

Extensions

Edited by Michael De Vlieger, Jan 23 2024