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A368471
a(n) is the sum of exponentially odd divisors of the largest unitary divisor of n that is an exponentially odd number (A268335).
2
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
1,2
LINKS
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... .
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]
PROG
(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)); }
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Dec 26 2023
STATUS
approved