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Union of prime powers and numbers that are not squarefree.
1

%I #15 Aug 03 2024 01:52:53

%S 1,2,3,4,5,7,8,9,11,12,13,16,17,18,19,20,23,24,25,27,28,29,31,32,36,

%T 37,40,41,43,44,45,47,48,49,50,52,53,54,56,59,60,61,63,64,67,68,71,72,

%U 73,75,76,79,80,81,83,84,88,89,90,92,96,97,98,99,100,101,103

%N Union of prime powers and numbers that are not squarefree.

%C Numbers that are prime powers p^m, m >= 0, or products of multiple powers of distinct primes p^m where at least 1 prime power p^m is such that m > 1.

%C Let N = A000027. Analogous to the following sequences:

%C A002808 = N \ {{1} U A000040} = {1} U A024619 U A013929,

%C A303554 = N \ A126706 = A000961 U A005117, and

%C A085961 = N \ {{1} U A246547} = {A005117 U A024619} \ {1}.

%H Michael De Vlieger, <a href="/A363597/b363597.txt">Table of n, a(n) for n = 1..10000</a>

%F Complement of A120944, i.e., A000027 \ A120944.

%F Union of A000961 and A013929.

%F Union of {1}, A000040, A126706, and A246547.

%e 1 is in the sequence because it is the empty product.

%e Prime p is in the sequence because it is not a composite squarefree number.

%e Numbers k that have prime power factors p^m | k where at least one prime power factor is such that m > 1 are in the sequence because they are not squarefree composites. Examples include 8, 9, 12, 20, and 36.

%t Select[Range[103], Nand[SquareFreeQ[#], CompositeQ[#]] &]

%o (PARI) isok(k) = (k==1) || isprimepower(k) || !issquarefree(k); \\ _Michel Marcus_, Aug 24 2023

%o (Python)

%o from math import isqrt

%o from sympy import mobius, primepi

%o def A363597(n):

%o if n==1: return 1

%o def f(x): return n-1+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))-primepi(x)

%o m, k = n-1, f(n-1)

%o while m != k:

%o m, k = k, f(k)

%o return m # _Chai Wah Wu_, Aug 02 2024

%Y Cf. A000040, A000961, A013929, A120944, A126706, A246547.

%K nonn,easy

%O 1,2

%A _Michael De Vlieger_, Aug 15 2023