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.

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

Author

Antti Karttunen, Nov 13 2021

Keywords

• nonn
• mult

Comments

Multiplicative because A349128 is.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A000010, A003961, A064216, A064989, A151799, A285702, A349127, A349128, A349138.

Programs

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 *)

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

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