A349122 - OEIS

%I #22 Dec 24 2022 03:50:25

%S 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,

%T 12,29,6,14,26,15,12,31,34,22,12,37,20,41,21,12,38,43,10,25,18,26,33,

%U 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

%N 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.

%C Multiplicative because A349128 is.

%H Antti Karttunen, <a href="/A349122/b349122.txt">Table of n, a(n) for n = 1..20000</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%F a(n) = Sum_{d|n} A349128(d).

%F For all n >= 1, a(A003961(n)) = n, a(2*n-1) = A064216(n).

%F From _Sebastian Karlsson_, Nov 15 2021: (Start)

%F a(2*n-1) = A064989(2*n-1).

%F Multiplicative with a(2^e) = e + 1 and a(p^e) = prevprime(p)^e for odd primes p. (End)

%F 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

%t 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 *)

%o (PARI)

%o 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)))); };

%o A349122(n) = sumdiv(n,d,A349128(d));

%o (Python)

%o from sympy import prevprime, factorint, prod

%o def f(p, e):

%o     return e+1 if p == 2 else prevprime(p)**e

%o def a(n):

%o     return prod(f(p, e) for p, e in factorint(n).items()) # _Sebastian Karlsson_, Nov 15 2021

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

%K nonn,mult

%O 1,2

%A _Antti Karttunen_, Nov 13 2021