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A349122
Inverse Möbius transform of A349128, where A349128(n) = phi(A064989(n)), A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and phi is Euler totient function.
3
1, 2, 2, 3, 3, 4, 5, 4, 4, 6, 7, 6, 11, 10, 6, 5, 13, 8, 17, 9, 10, 14, 19, 8, 9, 22, 8, 15, 23, 12, 29, 6, 14, 26, 15, 12, 31, 34, 22, 12, 37, 20, 41, 21, 12, 38, 43, 10, 25, 18, 26, 33, 47, 16, 21, 20, 34, 46, 53, 18, 59, 58, 20, 7, 33, 28, 61, 39, 38, 30, 67, 16, 71, 62, 18, 51, 35, 44, 73, 15, 16, 74, 79, 30, 39
OFFSET
1,2
COMMENTS
Multiplicative because A349128 is.
FORMULA
a(n) = Sum_{d|n} A349128(d).
For all n >= 1, a(A003961(n)) = n, a(2*n-1) = A064216(n).
From Sebastian Karlsson, Nov 15 2021: (Start)
a(2*n-1) = A064989(2*n-1).
Multiplicative with a(2^e) = e + 1 and a(p^e) = prevprime(p)^e for odd primes p. (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = (4/9) * Product_{p prime > 2} ((p^2-p)/(p^2-prevprime(p))) = 0.2942719052..., where prevprime is A151799. - Amiram Eldar, Dec 24 2022
MATHEMATICA
f[p_, e_] := NextPrime[p, -1]^e; f[2, e_] := e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 24 2022 *)
PROG
(PARI)
A349128(n) = { my(f = factor(n), q); prod(i=1, #f~, if(2==f[i, 1], 1, q = precprime(f[i, 1]-1); (q-1)*(q^(f[i, 2]-1)))); };
A349122(n) = sumdiv(n, d, A349128(d));
(Python)
from sympy import prevprime, factorint, prod
def f(p, e):
return e+1 if p == 2 else prevprime(p)**e
def a(n):
return prod(f(p, e) for p, e in factorint(n).items()) # Sebastian Karlsson, Nov 15 2021
KEYWORD
nonn,mult
AUTHOR
Antti Karttunen, Nov 13 2021
STATUS
approved