cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A074135 Triangle read by rows: for 1 <= k < n, a(n, k) is the least positive integer not already used. a(n, n) is the least positive integer not already used that makes the row sum a multiple of n.

Original entry on oeis.org

1, 2, 4, 3, 5, 7, 6, 8, 9, 13, 10, 11, 12, 14, 18, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 40, 36, 37, 38, 39, 41, 42, 43, 44, 49, 45, 46, 47, 48, 50, 51, 52, 53, 54, 64, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 75, 66, 67, 68, 69, 70, 71
Offset: 1

Views

Author

Amarnath Murthy, Aug 27 2002

Keywords

Comments

Alternative (but equivalent) definition: In the following square array, numbers (not occurring earlier) are entered like this a(1, 1), a(1, 2), a(2, 1), a(3, 1), a(2, 2), a(1, 3), a(1, 4), a(2, 3), a(3, 2), a(4, 1), a(5, 1), a(4, 2), ... such that the n-th diagonal sum is a multiple of n. 1 2 7 6 18... 4 5 8 14... 3 9 12... 13 11... 10... ... sequence contains terms as they are entered.

Crossrefs

Cf. A074132, A074133, A074134, A074136.
Cf. A082017, A082018, A082019, A082020.

Extensions

Edited and extended by David Wasserman, Oct 31 2006
Further edited by N. J. A. Sloane Jan 17 2009 at the suggestion of R. J. Mathar

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.

Original entry on oeis.org

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

Views

Author

Amarnath Murthy, Aug 27 2002

Keywords

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

A082018 First column of square array T(n,k) with T(1,1) = 1 where antidiagonals are filled alternating upwards and downwards with the smallest number not already used such that the n-th antidiagonal sum is a multiple of n.

Original entry on oeis.org

1, 4, 3, 13, 10, 21, 22, 40, 36, 64, 55, 81, 78, 115, 105, 138, 136, 182, 171, 211, 210, 265, 253, 300, 301, 364, 351, 433, 406, 477, 465, 556, 528, 606, 595, 695, 666, 751, 741, 850, 820, 912, 903, 1021, 990, 1089, 1081, 1208, 1176, 1282, 1275, 1411, 1378
Offset: 1

Views

Author

Amarnath Murthy, Apr 05 2003

Keywords

Comments

This is the boustrophedon method of filling an array. Sums of antidiagonals of T are in A074132. Sums of antidiagonals of T divided by number of antidiagonals are in A074133. Diagonal of T is in A082019.

Examples

			T(n,k) begins:
1,   2,  7,  6, 18, 15, ...
4,   5,  8, 14, 16, 27, ...
3,   9, 12, 17, 26, 31, ...
13, 11, 19, 25, 32, 42, ...
10, 20, 24, 33, 41, 50, ...
21, 23, 34, 39, 51, 60, ...

Crossrefs

Cf. A074135, A082017, A082019, A082020, A082002, A082003, A082004, A082005.

Extensions

Edited and more terms from Alois P. Heinz, Oct 26 2011
Showing 1-3 of 3 results.

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