This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A080259 #45 Aug 07 2025 20:25:38
%S A080259 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 A080259 37,38,39,40,41,42,43,44,45,46,47,49,50,51,52,53,55,56,57,58,59,61,62,
%U A080259 63,65,66,67,68,69,70,71,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87
%N A080259 Numbers whose squarefree kernel is not a primorial number, i.e., A007947(a(n)) is not in A002110.
%C A080259 Complement to A055932.
%C A080259 From _Michael De Vlieger_, Feb 06 2024: (Start)
%C A080259 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 A080259 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 A080259 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 A080259 Michael De Vlieger, <a href="/A080259/b080259.txt">Table of n, a(n) for n = 1..10000</a>
%F A080259 {a(n)} = { k : A053669(k) < A006530(k) }. - _Michael De Vlieger_, Jan 23 2024
%e A080259 From _Michael De Vlieger_, Jan 23 2024: (Start)
%e A080259 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 A080259 2 is not in the sequence since its squarefree kernel is 2, the smallest prime, hence the same as A002110(1) = 2.
%e A080259 4 is not in the sequence since its squarefree kernel is 2 = A002110(1).
%e A080259 (End)
%t A080259 Select[Range[120], Nor[IntegerQ@ Log2[#], And[EvenQ[#], Union@ Differences@ PrimePi[FactorInteger[#][[All, 1]]] == {1}]] &] (* _Michael De Vlieger_, Jan 23 2024 *)
%o A080259 (PARI) is(n) = {my(f=factor(n)[,1]);n>1&&primepi(f[#f])>#f} \\ _David A. Corneth_, May 22 2016
%o A080259 (Python)
%o A080259 from itertools import count, islice
%o A080259 from sympy import primepi, primefactors
%o A080259 def A080259_gen(startvalue=2): # generator of terms >= startvalue
%o A080259 for k in count(max(startvalue,2)):
%o A080259 p = list(map(primepi,primefactors(k)))
%o A080259 if not(min(p)==1 and max(p)==len(p)):
%o A080259 yield k
%o A080259 A080259_list = list(islice(A080259_gen(),40)) # _Chai Wah Wu_, Aug 07 2025
%Y A080259 Cf. A002110, A006530, A007947, A053669, A055932.
%K A080259 nonn
%O A080259 1,1
%A A080259 _Labos Elemer_, Mar 19 2003
%E A080259 Edited by _Michael De Vlieger_, Jan 23 2024