A082222 -id:A082222

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A082218

Square array T(i,j) in which for every k, the k-th partial sums of every row and column are divisible by k. Array read by antidiagonals, alternating upwards and downwards. Each entry is the least number not already used that fits the divisibility requirement.

+10
6

1, 3, 5, 6, 7, 2, 10, 12, 8, 4, 9, 14, 13, 16, 19, 25, 15, 37, 21, 23, 11, 20, 17, 22, 29, 26, 35, 24, 28, 36, 18, 32, 38, 44, 40, 48, 31, 56, 33, 68, 43, 50, 39, 34, 41, 27, 47, 61, 53, 57, 45, 75, 85, 93, 55, 30, 49, 65, 63, 72, 67, 88, 69, 62, 73, 51, 81, 83, 80, 70, 128, 42

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

In the square array T shown above, numbers (not occurring earlier) are entered like this, T(1, 1), T(1, 2), T(2, 1), T(3, 1), T(2, 2), T(1, 3), T(1, 4), T(2, 3), T(3, 2), T(4, 1), T(5, 1), T(4, 2), ... in such a way that every n-th partial sum of a row or a column is a multiple of n.

T(i, j) must satisfy a congruence mod i and another congruence mod j. i and j are not always relatively prime, but this pair of congruences is always solvable. See the link for a proof. - David Wasserman, Aug 26 2004

LINKS

Table of n, a(n) for n=1..72.

David Wasserman, Proof of a claim, 2004.

EXAMPLE

Square array T(i,j) (with rows i >= 1 and columns j >= 1) begins

1, 3, 2, 10, 19, 25, ...

5, 7, 12, 16, 15, ...

6, 8, 13, 37, ...

4, 14, 21, ...

9, 23, ...

11, ...

...

From Petros Hadjicostas, Feb 25 2021: (Start)

We start with T(1,1) = 1.

T(1,2) = 3 because i = 1, j = 2, and 1 + 3 = 4 which is divisible by j = 2. (We rejected 2 because 1 + 2 = 3, which is not divisible by 2.)

T(2,1) = 5 because i = 2 > 1, j = 1, and 1 + 5 = 6, which is divisible by j = 2. (We rejected 2 because 1 + 2 = 3, which is not divisible by j = 2. For the same reason, we rejected 4 because 1 + 4 = 5.)

T(3,1) = 6 because i = 3 > 1, j = 1, and 1 + 5 + 6 = 12, which is divisible by i = 3. (We rejected 2 because 1 + 5 + 2 = 8, which is not divisible by i = 3. For the same reason, we rejected 4 because 1 + 5 + 4 = 10.)

T(2,2) = 7 because i = 2 = j, 5 + 7 = 12, which is divisible by i = 2, and 3 + 7 = 10, which is divisible by j = 2. (We rejected 2, because 5 + 2 = 7 is not divisible by i = 2. We also rejected 4 because 5 + 4 = 9 is not divisible by i = 2.)

T(1,3) = 2 because i = 1, j = 3, and 1 + 3 + 2 = 6, which is divisible by j = 3. (End)

PROG

(PARI) lista(nn) = { my(a=matrix(nn, nn)); S=Set();

for(s=2, nn+1, if(s%2, i0=1; i1=s-1; i2=1, i0=s-1; i1=1; i2=-1);

forstep(i=i0, i1, i2, j=s-i;

ii=sum(k=1, j-1, a[i, k]); jj=sum(k=1, i-1, a[k, j]);

c=chinese(Mod(ii, j), Mod(jj, i));

t=component(c, 1)-lift(c); while(setsearch(S, t), t+=component(c, 1));

a[i, j]=t; S=setunion(S, [t]);

print1(", ", sum(k=1, j, a[i, j])/j); ))} \\ This is a modification of Max Alekseyev's PARI program from A082219. - Petros Hadjicostas, Feb 25 2021

CROSSREFS

Cf. A082219, A082220, A082221, A082222, A082223.

KEYWORD

nonn,easy,tabl

AUTHOR

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Apr 09 2003

EXTENSIONS

Edited and extended by David Wasserman, Aug 26 2004

STATUS

approved

A082219

First row of A082218.

+10
5

1, 3, 2, 10, 19, 25, 24, 28, 41, 27, 51, 81, 78, 86, 124, 120, 147, 123, 188, 142, 192, 116, 258, 250, 314, 254, 320, 392, 470, 404, 453, 377, 490, 612, 533, 445, 718, 708, 812, 602, 784, 726, 791, 771, 928, 1002, 1032, 1158, 996, 972, 1149, 1023, 1365, 1239

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..54.

PROG

(PARI) lista(nn) = { my(a=matrix(nn, nn)); S=Set();

for(s=2, nn+1, if(s%2, i0=1; i1=s-1; i2=1, i0=s-1; i1=1; i2=-1);

forstep(i=i0, i1, i2, j=s-i;

ii=sum(k=1, j-1, a[i, k]); jj=sum(k=1, i-1, a[k, j]);

c=chinese(Mod(ii, j), Mod(jj, i));

t=component(c, 1)-lift(c); while(setsearch(S, t), t+=component(c, 1));

a[i, j]=t; S=setunion(S, [t]);

if(i==1, print1(", ", sum(k=1, j, a[i, j])/j) ); ))} \\ Max Alekseyev, Nov 10 2007 [Slightly edited by Petros Hadjicostas, Feb 25 2021]

CROSSREFS

Cf. A082218, A082220, A082221, A082222, A082223.

KEYWORD

nonn

AUTHOR

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Apr 09 2003

EXTENSIONS

More terms from Max Alekseyev, Jun 08 2007

Edited by N. J. A. Sloane, Jun 11 2007

Various sections edited by Petros Hadjicostas, Feb 25 2021

STATUS

approved

A082220

First column of array A082218.

+10
5

1, 5, 6, 4, 9, 11, 20, 48, 31, 55, 30, 92, 52, 112, 94, 150, 113, 193, 152, 242, 218, 232, 246, 308, 301, 317, 441, 353, 311, 413, 431, 385, 399, 515, 570, 592, 614, 674, 542, 558, 697, 761, 785, 853, 969, 907, 745, 1099, 837, 1057, 1085, 1009, 1245, 1439, 1477

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..55.

PROG

(PARI) lista(nn) = { my(a=matrix(nn, nn)); S=Set();

for(s=2, nn+1, if(s%2, i0=1; i1=s-1; i2=1, i0=s-1; i1=1; i2=-1);

forstep(i=i0, i1, i2, j=s-i;

ii=sum(k=1, j-1, a[i, k]); jj=sum(k=1, i-1, a[k, j]);

c=chinese(Mod(ii, j), Mod(jj, i));

t=component(c, 1)-lift(c); while(setsearch(S, t), t+=component(c, 1));

a[i, j]=t; S=setunion(S, [t]);

if(j==1, print1(", ", sum(k=1, j, a[i, j])/j) ); ))} \\ This is a modification of Max Alekseyev's PARI program from A082219. - Petros Hadjicostas, Feb 25 2021

CROSSREFS

Cf. A082218, A082219, A082221, A082222, A082223.

KEYWORD

nonn

AUTHOR

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Apr 09 2003

EXTENSIONS

More terms from Max Alekseyev, Jun 08 2007

Edited by N. J. A. Sloane, Jun 11 2007

Name edited by Petros Hadjicostas, Feb 25 2021

STATUS

approved

A082221

Main diagonal of array A082218.

+10
5

1, 7, 13, 29, 43, 67, 82, 100, 139, 149, 240, 160, 270, 340, 373, 427, 426, 438, 579, 793, 505, 679, 877, 1037, 820, 994, 1240, 1268, 1221, 1109, 1369, 1487, 1800, 1678, 1745, 1607, 2406, 2042, 2560, 2356, 2701, 2407, 2625, 3127, 2791, 2971, 3210, 3170, 3642

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..49.

Tanya Khovanova, Non Recursions.

PROG

(PARI) lista(nn) = { my(a=matrix(nn, nn)); S=Set();

for(s=2, nn+1, if(s%2, i0=1; i1=s-1; i2=1, i0=s-1; i1=1; i2=-1);

forstep(i=i0, i1, i2, j=s-i;

ii=sum(k=1, j-1, a[i, k]); jj=sum(k=1, i-1, a[k, j]);

c=chinese(Mod(ii, j), Mod(jj, i));

t=component(c, 1)-lift(c); while(setsearch(S, t), t+=component(c, 1));

a[i, j]=t; S=setunion(S, [t]);

if(i==j, print1(", ", sum(k=1, j, a[i, j])/j) ); ))} \\ This is a modification of Max Alekseyev's PARI program from A082219. - Petros Hadjicostas, Feb 25 2021

CROSSREFS

Cf. A082218, A082219, A082220, A082222, A082223.

KEYWORD

nonn

AUTHOR

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Apr 09 2003

EXTENSIONS

More terms from Max Alekseyev, Jun 08 2007

Edited by N. J. A. Sloane, Jun 11 2007

Name edited by Petros Hadjicostas, Feb 25 2021

STATUS

approved

A082223

a(n) = (n-th partial sum of A082220)/n = (n-th partial sum of the first column of array A082218)/n.

+10
5

1, 3, 4, 4, 5, 6, 8, 13, 15, 19, 20, 26, 28, 34, 38, 45, 49, 57, 62, 71, 78, 85, 92, 101, 109, 117, 129, 137, 143, 152, 161, 168, 175, 185, 196, 207, 218, 230, 238, 246, 257, 269, 281, 294, 309, 322, 331, 347, 357, 371, 385, 397, 413, 432, 451, 468, 484, 504, 518

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..59.

PROG

(PARI) lista(nn) = { my(a=matrix(nn, nn)); my(b=vector(nn)); my(d=vector(nn)); S=Set(); for(s=2, nn+1, if(s%2, i0=1; i1=s-1; i2=1, i0=s-1; i1=1; i2=-1);

forstep(i=i0, i1, i2, j=s-i; ii=sum(k=1, j-1, a[i, k]); jj=sum(k=1, i-1, a[k, j]);

c=chinese(Mod(ii, j), Mod(jj, i));

t=component(c, 1)-lift(c); while(setsearch(S, t), t+=component(c, 1));

a[i, j]=t; S=setunion(S, [t]); b[i] = a[i, 1]; ); ); d[1]=b[1]; for(kk=1, nn, d[kk]=sum(ss=1, kk, b[ss])/kk); d; } \\ This is a modification of Max Alekseyev's PARI program from A082219. - Petros Hadjicostas, Feb 25 2021

CROSSREFS

Cf. A082218, A082219, A082220, A082221, A082222.

KEYWORD

nonn

AUTHOR

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Apr 09 2003

EXTENSIONS

More terms from Max Alekseyev, Jun 08 2007

Edited by N. J. A. Sloane, Jun 11 2007

Name edited by Petros Hadjicostas, Feb 25 2021

STATUS

approved

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