A162907 Sum of all numbers from tau(n) to sigma(n).

1, 5, 9, 25, 20, 72, 35, 114, 88, 165, 77, 391, 104, 294, 294, 486, 170, 765, 209, 888, 522, 660, 299, 1802, 493, 897, 814, 1581, 464, 2600, 527, 2001, 1170, 1479, 1170, 4150, 740, 1824, 1590, 4067, 902, 4628, 989, 3555, 3066, 2622, 1175, 7705, 1650, 4356, 2622, 4836

Offset

1,2

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Formula

a(n) = A000005(n) + (A000005(n)+1) + ... + A000203(n) = A000217(A000203(n)) - A000217(A000005(n)-1).

Example

From Antti Karttunen, Sep 10 2017: (Start)
For n=1, tau(1) = sigma(1) = 1, thus a(1) = 1.
For n=2, tau(2) = 2 & sigma(2) = 3, thus a(2) = 2 + 3 = 5
For n=3, tau(3) = 2 & sigma(3) = 4, thus a(3) = 2 + 3 + 4 = 9.
For n=4, tau(4) = 3 & sigma(4) = 7, thus a(4) = 3 + 4 + 5 + 6 + 7 = 25. (End)

Maple

A000217 := proc(n) n*(n+1) /2; end: A162907 := proc(n) A000217(numtheory[sigma](n)) - A000217(numtheory[tau](n)-1) ; end: seq(A162907(n), n=1..100) ; # R. J. Mathar, Jul 21 2009
# second Maple program:
with(numtheory):
a:= n-> ((u, o)-> (u+o)*(o-u+1)/2)(tau(n), sigma(n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Sep 10 2017

Mathematica

a[n_] := Range[DivisorSigma[0, n], DivisorSigma[1, n]] // Total;
Array[a, 100] (* Jean-François Alcover, Nov 20 2020 *)

Prog

(PARI)
A000217(n) = n * (n + 1) / 2;
A162907(n) = (A000217(sigma(n)) - A000217(numdiv(n)-1)); \\ Antti Karttunen, Sep 10 2017, after the Maple-code.

Crossrefs

Cf. A000005, A000203, A000217.

Sequence in context: A132354 A146419 A365070 * A165345 A177240 A074741

Adjacent sequences: A162904 A162905 A162906 * A162908 A162909 A162910

Keyword

nonn

Author

Juri-Stepan Gerasimov, Jul 17 2009

Extensions

Approximately half of the entries corrected by R. J. Mathar, Jul 21 2009

Status

approved