A359593 Multiplicative with a(p^e) = 1 if p divides e, p^e otherwise.
1, 2, 3, 1, 5, 6, 7, 8, 9, 10, 11, 3, 13, 14, 15, 1, 17, 18, 19, 5, 21, 22, 23, 24, 25, 26, 1, 7, 29, 30, 31, 32, 33, 34, 35, 9, 37, 38, 39, 40, 41, 42, 43, 11, 45, 46, 47, 3, 49, 50, 51, 13, 53, 2, 55, 56, 57, 58, 59, 15, 61, 62, 63, 1, 65, 66, 67, 17, 69, 70, 71, 72, 73, 74, 75, 19, 77, 78, 79, 5, 81, 82, 83, 21
Offset: 1
Links
- Antti Karttunen, Table of n, a(n) for n = 1..16384
Programs
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Mathematica
f[p_, e_] := If[Divisible[e, p], 1, p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 09 2023 *)
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PARI
A359593(n) = { my(f = factor(n)); prod(k=1, #f~, f[k, 1]^(f[k,2]*!!(f[k, 2]%f[k, 1]))); };
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Python
from math import prod from sympy import factorint def A359593(n): return prod(p**e for p, e in factorint(n).items() if e%p) # Chai Wah Wu, Jan 10 2023
Formula
a(n) = n / A359594(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 - p^(p-1)*(p-1)/(p^(2*p)-1)) = 0.4225104173... . - Amiram Eldar, Jan 11 2023
Comments