A019445 Form a permutation of the positive integers, p_1, p_2, ..., such that the average of each initial segment is an integer, using the greedy algorithm to define p_n; sequence gives p_1 + ... + p_n.

1, 4, 6, 12, 20, 24, 35, 40, 54, 70, 77, 96, 117, 126, 150, 160, 187, 216, 228, 260, 273, 308, 345, 360, 400, 442, 459, 504, 522, 570, 620, 640, 693, 748, 770, 828, 851, 912, 975, 1000, 1066, 1092, 1161, 1232, 1260, 1334, 1410, 1440, 1519, 1550

Offset: 1

Author

R. K. Guy, Tom Halverson (halverson(AT)macalester.edu)

Keywords

• nonn

Comments

It appears that a(n) is divisible by n. - Michael Somos, Jan 29 2004

Somos's conjecture is proved in both Shapovalov (1996) and Venkatachala (2009). - Jeffrey Shallit, Jul 18 2023

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• J. Shallit, Proving properties of some greedily-defined integer recurrences via automata theory, arXiv:2308.06544 [cs.DM], August 12 2023.
• A. Shapovalov, Problem M1517 (in Russian), Kvant 5 (1995), 20-21. English translation appeared in Quantum problem M185, Sept/October 1996 (beware, file is 75Mb).
• The Math Forum, Problem of the Week 818.
• B. J. Venkatachala, A curious bijection on natural numbers, JIS 12 (2009) 09.8.1.

Crossrefs

Cf. A019444, A019446.

Formula

Partial sums of A019444. - Sean A. Irvine, Mar 17 2019

a(n) = n * A019446(n). - Joerg Arndt, Jul 23 2023