A331737 Multiplicative with a(p^e) = p^A000265(e), where A000265(x) gives the odd part of x.
1, 2, 3, 2, 5, 6, 7, 8, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 24, 5, 26, 27, 14, 29, 30, 31, 32, 33, 34, 35, 6, 37, 38, 39, 40, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 54, 55, 56, 57, 58, 59, 30, 61, 62, 21, 8, 65, 66, 67, 34, 69, 70, 71, 24, 73, 74, 15, 38, 77, 78, 79, 10, 3, 82, 83, 42, 85, 86, 87, 88, 89, 30
Offset: 1
Links
- Antti Karttunen, Table of n, a(n) for n = 1..16384
- Brahim Mittou, New properties of an arithmetic function, Mathematica Montisnigri, Vol LIII (2022).
Programs
-
Mathematica
f[p_, e_] := p^(e/2^IntegerExponent[e, 2]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 17 2022 *)
-
PARI
A000265(n) = (n>>valuation(n,2)); A331737(n) = { my(f = factor(n)); prod(k=1, #f~, f[k, 1]^A000265(f[k, 2])); };
Formula
a(n) = n / A331738(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((1-1/p) * Sum_{k>=1} p^(2^k - 1)/(p^(2^(k+1)-2) - 1)) = 0.3953728204... . - Amiram Eldar, Nov 17 2022
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