A366535 - OEIS

%I #8 Oct 12 2023 09:51:31

%S 1,3,4,6,12,8,9,18,12,14,24,24,18,20,32,36,24,36,42,28,30,72,32,33,48,

%T 54,48,38,60,56,54,42,96,44,72,48,72,54,84,72,72,80,90,60,62,96,84,

%U 144,68,96,144,72,74,114,96,168,80,126,84,108,132,120,108,90,112

%N The sum of unitary divisors of the exponentially odd numbers (A268335).

%H Amiram Eldar, <a href="/A366535/b366535.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A034448(A268335(n)).

%F Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = (zeta(4)/d^2) * Product_{p prime} (1 - 2/p^4 + 1/p^5) = 1.92835521961603199612..., d = A065463 is the asymptotic density of the exponentially odd numbers.

%F The asymptotic mean of the unitary abundancy index of the exponentially odd numbers: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A268335(k) = c * d = 1.35841479521454692063... .

%t s[n_] := Module[{f = FactorInteger[n], e}, e = f[[;;, 2]]; If[AllTrue[e, OddQ], Times @@ (1 + Power @@@ f), Nothing]]; s[1] = 1; Array[s, 100]

%o (PARI) lista(max) = for(k = 1, max, my(f = factor(k), e = f[, 2], isexpodd = 1); for(i = 1, #e, if(!(e[i] % 2), isexpodd = 0; break)); if(isexpodd, print1(prod(i = 1, #e, 1 + f[i, 1]^e[i]), ", ")));

%Y Cf. A034448, A065463, A077610, A268335, A366439, A366534.

%Y Similar sequences: A034676, A366537, A366539.

%K nonn,easy

%O 1,2

%A _Amiram Eldar_, Oct 12 2023