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A363597
Union of prime powers and numbers that are not squarefree.
1
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, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 84, 88, 89, 90, 92, 96, 97, 98, 99, 100, 101, 103
OFFSET
1,2
COMMENTS
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.
Let N = A000027. Analogous to the following sequences:
A002808 = N \ {{1} U A000040} = {1} U A024619 U A013929,
A085961 = N \ {{1} U A246547} = {A005117 U A024619} \ {1}.
LINKS
FORMULA
Complement of A120944, i.e., A000027 \ A120944.
Union of A000961 and A013929.
Union of {1}, A000040, A126706, and A246547.
EXAMPLE
1 is in the sequence because it is the empty product.
Prime p is in the sequence because it is not a composite squarefree number.
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.
MATHEMATICA
Select[Range[103], Nand[SquareFreeQ[#], CompositeQ[#]] &]
PROG
(PARI) isok(k) = (k==1) || isprimepower(k) || !issquarefree(k); \\ Michel Marcus, Aug 24 2023
(Python)
from math import isqrt
from sympy import mobius, primepi
def A363597(n):
if n==1: return 1
def f(x): return n-1+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))-primepi(x)
m, k = n-1, f(n-1)
while m != k:
m, k = k, f(k)
return m # Chai Wah Wu, Aug 02 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Michael De Vlieger, Aug 15 2023
STATUS
approved