A368471 a(n) is the sum of exponentially odd divisors of the largest unitary divisor of n that is an exponentially odd number (A268335).

1, 3, 4, 1, 6, 12, 8, 11, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 44, 1, 42, 31, 8, 30, 72, 32, 43, 48, 54, 48, 1, 38, 60, 56, 66, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 93, 72, 88, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144

Offset: 1

Author

Amiram Eldar, Dec 26 2023

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

Cf. A000203, A000290, A005117, A033634, A268335, A350389, A368470.

Programs

Mathematica

f[p_, e_] := If[OddQ[e], 1 + (p^(e + 2) - p)/(p^2 - 1), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2]%2, (f[i,1]^(f[i,2]+2) - f[i,1])/(f[i,1]^2 - 1) + 1, 1));}

Formula

a(n) = A033634(A350389(n)).

Multiplicative with a(p^e) = (p^(e+2) - p)/(p^2 - 1) + 1 if e is odd and 1 otherwise.

a(n) >= 1, with equality if and only if n is a square (A000290).

a(n) <= A000203(n), with equality if and only if n is squarefree (A005117).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^6/1080) * Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 0.51287686448947428073... .