A259823 -id:A259823 - cp's OEIS frontend

A174030 Partial sums of A007694.

1, 3, 7, 13, 21, 33, 49, 67, 91, 123, 159, 207, 261, 325, 397, 493, 601, 729, 873, 1035, 1227, 1443, 1699, 1987, 2311, 2695, 3127, 3613, 4125, 4701, 5349, 6117, 6981, 7953, 8977, 10129, 11425, 12883, 14419, 16147, 18091, 20139, 22443, 25035, 27951, 31023

Offset: 1

Jonathan Vos Post, Mar 06 2010

nonn

Partial sums of numbers k such that phi(k) divides k.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000010, A007694, A259823.

Accumulate[Select[Range[5000], Divisible[#, EulerPhi[#]] &]] (* Amiram Eldar, Nov 05 2024 *)

a(n) = Sum_{i=1..n} A007694(i).

a(n) = 2*A259823(n-1) + 1. - Amiram Eldar, Nov 05 2024

A291876 Consider the graph with one central vertex connected to three outer vertices (a star graph). Then, a(n) is the minimum number of moves required to transfer a stack of n pegs from one outer vertex to another outer vertex, moving pegs to adjacent vertices, following the rules of the Towers of Hanoi.

Original entry on oeis.org

2, 6, 12, 20, 32, 48, 66, 90, 122, 158, 206, 260, 324, 396, 492, 600, 728, 872, 1034, 1226, 1442, 1698, 1986, 2310, 2694, 3126, 3612, 4124, 4700, 5348, 6116, 6980, 7952, 8976, 10128, 11424, 12882, 14418, 16146, 18090, 20138, 22442, 25034, 27950, 31022, 34478, 38366

Offset: 1

Eric M. Schmidt, Sep 04 2017

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Thierry Bousch, La Tour de Stockmeyer, Séminaire Lotharingien de Combinatoire 77 (2017), Article B77d.
Caroline Holz auf der Heide, Distances and automatic sequences in distinguished variants of Hanoi graphs, Dissertation. Fakultät für Mathematik, Informatik und Statistik. Ludwig-Maximilians-Universität München, 2016. [See Chapter 3.]
Paul K. Stockmeyer, Variations on the Four-Post Tower of Hanoi Puzzle, Congr. Numer., 102 (1994), pp. 3-12.
Eric Weisstein's World of Mathematics, Star Graph
Index entries for sequences related to Towers of Hanoi

Cf. A259823, A291877.

A[0]:= 0:
A[1]:= 2:
for n from 2 to 100 do A[n]:= min(seq(3*A[k]+2^(n-k+1)-2, k=0..n-1)) od:
seq(A[i],i=1..100); # Robert Israel, Oct 27 2017

Conjecturally, a(n) = 2*A259823(n).

This conjecture was proved by Thierry Bousch, see link. - Paul Zimmermann, Oct 05 2015

Terms a(17) and beyond from Robert Israel, Oct 27 2017

cp's OEIS Frontend

A174030 Partial sums of A007694.

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