A169999 -id:A169999

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A325820

Multiplication table for carryless product i X j in base 3 for i >= 0 and j >= 0, read by antidiagonals.

+10
11

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 1, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 8, 9, 8, 5, 0, 0, 6, 7, 12, 12, 7, 6, 0, 0, 7, 3, 15, 16, 15, 3, 7, 0, 0, 8, 5, 18, 11, 11, 18, 5, 8, 0, 0, 9, 4, 21, 24, 13, 24, 21, 4, 9, 0, 0, 10, 18, 24, 19, 21, 21, 19, 24, 18, 10, 0, 0, 11, 20, 27, 23, 26, 9, 26, 23, 27, 20, 11, 0, 0, 12, 19, 30, 36, 19, 15, 15, 19, 36, 30, 19, 12, 0

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

OFFSET

0,8

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..10584; the first 145 antidiagonals of array

Index entries for sequences related to carryless arithmetic

EXAMPLE

The array begins as:

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...

0, 2, 1, 6, 8, 7, 3, 5, 4, 18, 20, 19, 24, ...

0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, ...

0, 4, 8, 12, 16, 11, 24, 19, 23, 36, 40, 44, 48, ...

0, 5, 7, 15, 11, 13, 21, 26, 19, 45, 50, 52, 33, ...

0, 6, 3, 18, 24, 21, 9, 15, 12, 54, 60, 57, 72, ...

0, 7, 5, 21, 19, 26, 15, 13, 11, 63, 70, 68, 57, ...

0, 8, 4, 24, 23, 19, 12, 11, 16, 72, 80, 76, 69, ...

0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, ...

0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 83, 120, ...

0, 11, 19, 33, 44, 52, 57, 68, 76, 99, 83, 91, 132, ...

0, 12, 24, 36, 48, 33, 72, 57, 69, 108, 120, 132, 144, ...

etc.

A(2,2) = 2*2 mod 3 = 1.

PROG

(PARI)

up_to = 105;

A325820sq(b, c) = fromdigits(Vec(Pol(digits(b, 3))*Pol(digits(c, 3)))%3, 3);

A325820list(up_to) = { my(v = vector(up_to), i=0); for(a=0, oo, for(col=0, a, if(i++ > up_to, return(v)); v[i] = A325820sq(a-col, col))); (v); };

v325820 = A325820list(up_to);

A325820(n) = v325820[1+n];

CROSSREFS

Cf. A169999 (the main diagonal).

Row/Column 0: A000004, Row/Column 1: A001477, Row/Column 2: A004488, Row/Column 3: A008585, Row/Column 4: A242399, Row/Column 9: A008591.

Cf. A007089, A207669, A207670, A263273, A325825.

Cf. A325821 (same table without the zero row and column).

Cf. A048720 (binary), A059692 (decimal), A004247 (full multiply).

KEYWORD

nonn,base,tabl

AUTHOR

Antti Karttunen, May 22 2019

STATUS

approved

A170990

Carryless product n X n in base 9.

+10
8

0, 1, 4, 0, 7, 7, 0, 4, 1, 81, 100, 121, 135, 160, 97, 108, 130, 145, 324, 361, 400, 351, 394, 349, 378, 337, 370, 0, 55, 31, 0, 61, 34, 0, 58, 28, 567, 640, 634, 621, 619, 610, 594, 589, 577, 567, 577, 589, 594, 610, 619, 621, 634, 640, 0, 28, 58, 0, 34, 61, 0, 31, 55, 324, 370

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

OFFSET

0,3

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..6560

David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.

Index entries for sequences related to carryless arithmetic

PROG

(PARI) a(n) = fromdigits(Vec(Pol(digits(n, 9))^2)%9, 9); \\ Seiichi Manyama, Mar 09 2023

CROSSREFS

For bases 2 through 10 see A000695, A169999, A170985-A170990 and A059729. A169821 is a less satisfactory variant.

KEYWORD

nonn,base,look

AUTHOR

N. J. A. Sloane, Aug 29 2010

STATUS

approved

A170985

Carryless product n X n in base 4.

+10
7

0, 1, 0, 1, 16, 25, 16, 25, 0, 1, 0, 1, 16, 25, 16, 25, 256, 289, 256, 289, 400, 441, 400, 441, 256, 289, 256, 289, 400, 441, 400, 441, 0, 1, 0, 1, 16, 25, 16, 25, 0, 1, 0, 1, 16, 25, 16, 25, 256, 289, 256, 289, 400, 441, 400, 441, 256, 289, 256, 289, 400, 441, 400, 441, 4096, 4225

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

OFFSET

0,5

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..4095

David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.

Index entries for sequences related to carryless arithmetic

PROG

(PARI) a(n) = fromdigits(Vec(Pol(digits(n, 4))^2)%4, 4); \\ Seiichi Manyama, Mar 09 2023

CROSSREFS

For bases 2 through 10 see A000695, A169999, A170985-A170990 and A059729.

KEYWORD

nonn,base,look

AUTHOR

N. J. A. Sloane, Aug 29 2010

STATUS

approved

A170986

Carryless product n X n in base 5.

+10
1

0, 1, 4, 4, 1, 25, 36, 49, 34, 41, 100, 121, 119, 114, 106, 100, 106, 114, 119, 121, 25, 41, 34, 49, 36, 625, 676, 729, 654, 701, 900, 961, 899, 934, 991, 1225, 1171, 1219, 1139, 1181, 850, 781, 839, 769, 821, 1025, 1091, 1009, 1074, 1111, 2500, 2601, 2579, 2554, 2526

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

OFFSET

0,3

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..3124

David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.

Index entries for sequences related to carryless arithmetic

PROG

(PARI) a(n) = fromdigits(Vec(Pol(digits(n, 5))^2)%5, 5); \\ Seiichi Manyama, Mar 09 2023

CROSSREFS

For bases 2 through 10 see A000695, A169999, A170985-A170990 and A059729.

KEYWORD

nonn,base,look

AUTHOR

N. J. A. Sloane, Aug 29 2010

STATUS

approved

A170987

Carryless product n X n in base 6.

+10
1

0, 1, 4, 3, 4, 1, 36, 49, 64, 39, 52, 61, 144, 169, 160, 147, 172, 157, 108, 109, 112, 111, 112, 109, 144, 157, 172, 147, 160, 169, 36, 61, 52, 39, 64, 49, 1296, 1369, 1444, 1299, 1372, 1441, 1764, 1849, 1936, 1767, 1852, 1933, 2304, 2185, 2248, 2307, 2188, 2245

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

OFFSET

0,3

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..7775

David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.

Index entries for sequences related to carryless arithmetic

PROG

(PARI) a(n) = fromdigits(Vec(Pol(digits(n, 6))^2)%6, 6); \\ Seiichi Manyama, Mar 09 2023

CROSSREFS

For bases 2 through 10 see A000695, A169999, A170985-A170990 and A059729.

KEYWORD

nonn,base,look

AUTHOR

N. J. A. Sloane, Aug 29 2010

STATUS

approved

A170988

Carryless product n X n in base 7.

+10
1

0, 1, 4, 2, 2, 4, 1, 49, 64, 81, 93, 58, 74, 85, 196, 225, 207, 233, 212, 242, 218, 98, 141, 137, 128, 121, 116, 106, 98, 106, 116, 121, 128, 137, 141, 196, 218, 242, 212, 233, 207, 225, 49, 85, 74, 58, 93, 81, 64, 2401, 2500, 2601, 2697, 2452, 2552, 2647, 3136, 3249

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

OFFSET

0,3

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..2400

David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.

Index entries for sequences related to carryless arithmetic

PROG

(PARI) a(n) = fromdigits(Vec(Pol(digits(n, 7))^2)%7, 7); \\ Seiichi Manyama, Mar 09 2023

CROSSREFS

For bases 2 through 10 see A000695, A169999, A170985-A170990 and A059729.

KEYWORD

nonn,base,look

AUTHOR

N. J. A. Sloane, Aug 29 2010

STATUS

approved

A170989

Carryless product n X n in base 8.

+10
1

0, 1, 4, 1, 0, 1, 4, 1, 64, 81, 100, 113, 64, 81, 100, 113, 256, 289, 260, 289, 256, 289, 260, 289, 64, 113, 100, 81, 64, 113, 100, 81, 0, 1, 4, 1, 0, 1, 4, 1, 64, 81, 100, 113, 64, 81, 100, 113, 256, 289, 260, 289, 256, 289, 260, 289, 64, 113, 100, 81, 64, 113, 100, 81, 4096, 4225

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

OFFSET

0,3

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..4095

David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.

Index entries for sequences related to carryless arithmetic

PROG

(PARI) a(n) = fromdigits(Vec(Pol(digits(n, 8))^2)%8, 8); \\ Seiichi Manyama, Mar 09 2023

CROSSREFS

For bases 2 through 10 see A000695, A169999, A170985-A170990 and A059729.

KEYWORD