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

1,1

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)

Links

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

Formula

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

Example

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)

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A326980 A281005 A379851 * A067715 A344291 A231564

Adjacent sequences: A080256 A080257 A080258 * A080260 A080261 A080262

Keyword

nonn

Author

Labos Elemer, Mar 19 2003

Extensions

Edited by Michael De Vlieger, Jan 23 2024

Status

approved