

A074133


Average of the nth group, if positive integers are rearranged in groups of k=1,2,3,... numbers whose sum is a multiple of k.


3



1, 3, 5, 9, 13, 18, 25, 33, 41, 51, 61, 72, 85, 99, 113, 128, 145, 163, 181, 200, 221, 243, 265, 288, 313, 339, 365, 393, 421, 450, 481, 513, 545, 578, 613, 649, 685, 722, 761, 801, 841, 882, 925, 969, 1013, 1058, 1105, 1153, 1201, 1250, 1301, 1353, 1405
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OFFSET

1,2


COMMENTS

The lexicographically earliest possibility satisfying the requirement is to be considered. This practically means that the kth group consists of the k1 smallest numbers not yet used, followed the next smallest unused number which completes their sum to a multiple of k.  M. F. Hasler, May 07 2010
Original definition (The given example seems wrong from "23" on. [Z. Seidov & MFH]): Rearrange the natural numbers in groups so that the nth group contains n terms and the sum is a multiple of n: (1), (2, 4), (3, 5, 7), (6, 8, 9, 13), (10, 11, 12, 14, 23), (15, 16, 17, 18, 19, 29), (20, 21, 22, 24, 25, 26, 30), (27, 28, 31, 32, 33, 34, 35, 36), ... Sequence gives sum of the terms/ n for the nth group (the average of the group).
In the kth group the first (k1) terms are partly all the numbers not included earlier followed by numbers in increasing order and then the kth term is chosen so that the sum is a multiple of k.


LINKS

Table of n, a(n) for n=1..53.


EXAMPLE

From M. F. Hasler, May 07 2010: (Start)
Below the groups are given inside /*...*/, followed by the average a(n):
/*[1]*/ 1,
/*[2, 4]*/ 3,
/*[3, 5, 7]*/ 5,
/*[6, 8, 9, 13]*/ 9,
/*[10, 11, 12, 14, 18]*/ 13,
/*[15, 16, 17, 19, 20, 21]*/ 18,
/*[22,23, 24, 25, 26, 27, 28]*/ 25,
/*[29, 30, 31, 32, 33, 34, 35, 40]*/ 33,
/*[36, 37, 38, 39, 41, 42, 43, 44, 49]*/ 41,
/*[45, 46, 47, 48, 50, 51, 52, 53, 54, 64]*/ 51,
/*[55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 75]*/ 61,
/*[66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 81]*/ 72, ... (End)


PROG

(PARI) {v=vector(10000); print1(v[1]=1); for(n=2, 80, s=0; i=1; for(k=1, n1, while(v[i], i++); s+=i; v[i]=1); i=ceil(s/n)*ns; while(! iv[i], i+=n); s+=i; v[i]=1; print1(", "s/n))}
(PARI)
/* Removing the code involving "show" considerably speeds up the computation */
A074133(n, show=0 /* 1:print terms 1..n, 2:print all groups */)={
my(t, u=0, lu=1 /* least unused */); for(k=1, n, my(v=[]); s=sum( i=1, k1,
t=lu; while(bittest(u, t), t++); show>1 & v=concat(v, t); u+=1<<t;
t==lu & while( bittest(u, lu++), ); t); forstep(n=max(lu1, s)\k*k+k, 1e9, k,
bittest(u, ns) & next; u+=1<<(ns); show>1 & print1("/*"concat(v, ns)"*/ ");
t=n/k; show & print1(t, ", "); break)); t} \\ M. F. Hasler, May 07 2010


CROSSREFS

Cf. A074132, A074134, A074135.
Sequence in context: A129230 A203567 A071404 * A215909 A078735 A212530
Adjacent sequences: A074130 A074131 A074132 * A074134 A074135 A074136


KEYWORD

nonn,changed


AUTHOR

Amarnath Murthy, Aug 27 2002


EXTENSIONS

Corrected and extended by Ralf Stephan, Mar 26 2003
Edited by M. F. Hasler, May 09 2010


STATUS

approved



