A336475 - OEIS

%I #28 Sep 21 2023 01:46:11

%S 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,

%T 108,14,58,60,62,1,132,34,140,27,74,38,156,10,82,84,86,22,270,46,94,6,

%U 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

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

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

%C 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.

%H Antti Karttunen, <a href="/A336475/b336475.txt">Table of n, a(n) for n = 1..16384</a>

%H Antti Karttunen, <a href="/A336475/a336475.txt">Data supplement: n, a(n) computed for n = 1..65537</a>

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

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

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

%F From _Amiram Eldar_, Sep 21 2023: (Start)

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

%F 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)

%t 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 *)

%o (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]))); };

%o (Python)

%o from sympy import divisor_count

%o def A336475(n): return (m:=n>>(~n&n-1).bit_length())*divisor_count(m) # _Chai Wah Wu_, Jul 13 2022

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

%K nonn,easy,mult

%O 1,3

%A _Antti Karttunen_, Jul 30 2020