A074133 Average of the n-th group, if positive integers are rearranged in groups of k=1,2,3,... numbers whose sum is a multiple of k.

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

Offset: 1

Author

Amarnath Murthy, Aug 27 2002

Keywords

• nonn

Comments

The lexicographically earliest possibility satisfying the requirement is to be considered. This practically means that the k-th group consists of the k-1 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 n-th 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 n-th group (the average of the group).

In the k-th group the first (k-1) terms are partly all the numbers not included earlier followed by numbers in increasing order and then the k-th term is chosen so that the sum is a multiple of k.

Examples

			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)
		

Crossrefs

Cf. A074132, A074134, A074135.

Programs

PARI

{v=vector(10000); print1(v[1]=1); for(n=2,80, s=0;i=1; for(k=1,n-1, while(v[i],i++); s+=i; v[i]=1); i=ceil(s/n)*n-s; while(! i||v[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,k-1,
t=lu; while(bittest(u,t), t++); show>1 & v=concat(v,t); u+=1<1 & print1("/*"concat(v,n-s)"*/ ");
t=n/k; show & print1(t,", "); break));t} \\ M. F. Hasler, May 07 2010

Formula

a(n) = A074132(n) / n. - Sean A. Irvine, Jan 12 2025

Extensions

Corrected and extended by Ralf Stephan, Mar 26 2003

Edited by M. F. Hasler, May 09 2010