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 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. 3

%I

%S 1,3,5,9,13,18,25,33,41,51,61,72,85,99,113,128,145,163,181,200,221,

%T 243,265,288,313,339,365,393,421,450,481,513,545,578,613,649,685,722,

%U 761,801,841,882,925,969,1013,1058,1105,1153,1201,1250,1301,1353,1405

%N 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.

%C 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

%C 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).

%C 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.

%e From _M. F. Hasler_, May 07 2010: (Start)

%e Below the groups are given inside /*...*/, followed by the average a(n):

%e /*[1]*/ 1,

%e /*[2, 4]*/ 3,

%e /*[3, 5, 7]*/ 5,

%e /*[6, 8, 9, 13]*/ 9,

%e /*[10, 11, 12, 14, 18]*/ 13,

%e /*[15, 16, 17, 19, 20, 21]*/ 18,

%e /*[22,23, 24, 25, 26, 27, 28]*/ 25,

%e /*[29, 30, 31, 32, 33, 34, 35, 40]*/ 33,

%e /*[36, 37, 38, 39, 41, 42, 43, 44, 49]*/ 41,

%e /*[45, 46, 47, 48, 50, 51, 52, 53, 54, 64]*/ 51,

%e /*[55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 75]*/ 61,

%e /*[66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 81]*/ 72, ... (End)

%o (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))}

%o (PARI)

%o /* Removing the code involving "show" considerably speeds up the computation */

%o A074133(n,show=0 /* 1:print terms 1..n, 2:print all groups */)={

%o my(t, u=0, lu=1 /* least unused */); for(k=1, n, my(v=[]); s=sum( i=1,k-1,

%o t=lu; while(bittest(u,t), t++); show>1 & v=concat(v,t); u+=1<<t;

%o t==lu & while( bittest(u,lu++),); t); forstep(n=max(lu-1,s)\k*k+k,1e9,k,

%o bittest(u,n-s) & next; u+=1<<(n-s); show>1 & print1("/*"concat(v,n-s)"*/ ");

%o t=n/k; show & print1(t,", "); break));t} \\ _M. F. Hasler_, May 07 2010

%Y Cf. A074132, A074134, A074135.

%K nonn

%O 1,2

%A _Amarnath Murthy_, Aug 27 2002

%E Corrected and extended by _Ralf Stephan_, Mar 26 2003

%E Edited by _M. F. Hasler_, May 09 2010

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Last modified May 28 09:03 EDT 2022. Contains 354112 sequences. (Running on oeis4.)