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Numbers whose squarefree kernel is not a primorial number, i.e., A007947(a(n)) is not in A002110.
15

%I #40 Apr 28 2024 16:24:10

%S 3,5,7,9,10,11,13,14,15,17,19,20,21,22,23,25,26,27,28,29,31,33,34,35,

%T 37,38,39,40,41,42,43,44,45,46,47,49,50,51,52,53,55,56,57,58,59,61,62,

%U 63,65,66,67,68,69,70,71,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87

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

%C Complement to A055932.

%C From _Michael De Vlieger_, Feb 06 2024: (Start)

%C 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.

%C 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.

%C 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)

%H Michael De Vlieger, <a href="/A080259/b080259.txt">Table of n, a(n) for n = 1..10000</a>

%F {a(n)} = { k : A053669(k) < A006530(k) }. - _Michael De Vlieger_, Jan 23 2024

%e From _Michael De Vlieger_, Jan 23 2024: (Start)

%e 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.

%e 2 is not in the sequence since its squarefree kernel is 2, the smallest prime, hence the same as A002110(1) = 2.

%e 4 is not in the sequence since its squarefree kernel is 2 = A002110(1).

%e (End)

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

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

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

%K nonn

%O 1,1

%A _Labos Elemer_, Mar 19 2003

%E Edited by _Michael De Vlieger_, Jan 23 2024