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A344705
a(n) = n + A001615(n) - sigma(n), where A001615 is the Dedekind psi-function, and sigma(n) gives the sum of divisors of n; difference between psi and the sum of proper divisors.
5
1, 2, 3, 3, 5, 6, 7, 5, 8, 10, 11, 8, 13, 14, 15, 9, 17, 15, 19, 14, 21, 22, 23, 12, 24, 26, 23, 20, 29, 30, 31, 17, 33, 34, 35, 17, 37, 38, 39, 22, 41, 42, 43, 32, 39, 46, 47, 20, 48, 47, 51, 38, 53, 42, 55, 32, 57, 58, 59, 36, 61, 62, 55, 33, 65, 66, 67, 50, 69, 70, 71, 21, 73, 74, 71, 56, 77, 78, 79, 38, 68, 82
OFFSET
1,2
COMMENTS
First negative term occurs as a(1440) = -18.
FORMULA
a(n) = A001615(n) - A001065(n) = n - A244963(n) = n + A001615(n) - sigma(n).
a(n) = A033879(n) + A306927(n).
a(n) = n + A344753(n) - 2*A001065(n).
Sum_{k=1..n} a(k) = c * n^2 / 2 + O(n*log(n)), where c = 15/Pi^2 + 1 - Pi^2/6 = 0.874883... . - Amiram Eldar, Dec 08 2023
MATHEMATICA
f[p_, e_] := (p^(e+1) - 1)/(p-1); a[1] = 1; a[n_] := Module[{fct = FactorInteger[n]}, n * (Times @@ (1 + 1/fct[[;; , 1]]) + 1) - Times @@ f @@@ fct]; Array[a, 100] (* Amiram Eldar, Dec 08 2023 *)
PROG
(PARI)
A001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d));
A344705(n) = ((n + A001615(n)) - sigma(n));
CROSSREFS
Cf. A000203, A001065, A001615, A033879, A244963, A291209 (positions of zeros), A344700, A344704, A344753.
Sequence in context: A085310 A055653 A155918 * A331170 A325183 A343247
KEYWORD
sign,easy
AUTHOR
Antti Karttunen, May 28 2021
STATUS
approved