A336475 Multiplicative with a(2^e) = 1, and for odd primes p, a(p^e) = (e+1)*p^e.

1, 1, 6, 1, 10, 6, 14, 1, 27, 10, 22, 6, 26, 14, 60, 1, 34, 27, 38, 10, 84, 22, 46, 6, 75, 26, 108, 14, 58, 60, 62, 1, 132, 34, 140, 27, 74, 38, 156, 10, 82, 84, 86, 22, 270, 46, 94, 6, 147, 75, 204, 26, 106, 108, 220, 14, 228, 58, 118, 60, 122, 62, 378, 1, 260, 132, 134, 34, 276, 140, 142, 27, 146, 74, 450, 38, 308, 156

Offset: 1

Author

Antti Karttunen, Jul 30 2020

Keywords

• nonn
• easy
• mult

Comments

Dirichlet convolution of A000265 with itself, divided by A001511(n).

Although for all i, j: A003602(i) = A003602(j) => a(i) = a(j), it is not true that a(i) = a(j) => A003602(i) = A003602(j), because A038040 has a duplicate occurrence of a term on at least two odd positions: A038040(1875) = A038040(3125) = 18750.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Crossrefs

Cf. A000005, A000265, A001227, A001511, A001620, A003602, A038040, A336476.

Programs

Mathematica

f[p_, e_] := (e+1)*p^e; f[2, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 21 2023 *)

PARI

A336475(n) = { my(f=factor(n)); prod(i=1, #f~, if(2==f[i,1],1,(1+f[i,2]) * (f[i,1]^f[i,2]))); };

Python

from sympy import divisor_count
def A336475(n): return (m:=n>>(~n&n-1).bit_length())*divisor_count(m) # Chai Wah Wu, Jul 13 2022

Formula

a(n) = A038040(A000265(n)).

a(n) = A000265(n) * A001227(n).

a(n) = (A000005(n) * A000265(n)) / A001511(n). [See the first comment]

From Amiram Eldar, Sep 21 2023: (Start)

Dirichlet g.f.: ((2^s - 2)^2/(4^s - 2^s)) * zeta(s-1)^2.

Sum_{k=1..n} a(k) ~ (n^2/12) * (2*log(n) + 4*gamma + 10*log(2)/3 - 1), where gamma is Euler's constant (A001620). (End)