A366535 The sum of unitary divisors of the exponentially odd numbers (A268335).

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, 54, 48, 38, 60, 56, 54, 42, 96, 44, 72, 48, 72, 54, 84, 72, 72, 80, 90, 60, 62, 96, 84, 144, 68, 96, 144, 72, 74, 114, 96, 168, 80, 126, 84, 108, 132, 120, 108, 90, 112

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

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

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.

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... .

Mathematica

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]

Prog

(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]), ", ")));

Crossrefs

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

Similar sequences: A034676, A366537, A366539.

Sequence in context: A002090 A199970 A283738 * A368469 A374457 A366439

Adjacent sequences: A366532 A366533 A366534 * A366536 A366537 A366538

Keyword

nonn,easy

Author

Amiram Eldar, Oct 12 2023

Status

approved