A059729 -id:A059729 - cp's OEIS frontend

A359697 Triangle T(n,k), n >= 1, 1 <= k <= n, read by rows, where T(n,k) is carryless product n X k base 10.

1, 2, 4, 3, 6, 9, 4, 8, 2, 6, 5, 0, 5, 0, 5, 6, 2, 8, 4, 0, 6, 7, 4, 1, 8, 5, 2, 9, 8, 6, 4, 2, 0, 8, 6, 4, 9, 8, 7, 6, 5, 4, 3, 2, 1, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 12, 24, 36, 48, 50, 62, 74, 86, 98, 120, 132, 144

Offset: 1

Seiichi Manyama, Mar 08 2023

			Triangle begins:
   1;
   2,  4;
   3,  6,  9;
   4,  8,  2,  6;
   5,  0,  5,  0,  5;
   6,  2,  8,  4,  0,  6;
   7,  4,  1,  8,  5,  2,  9;
   8,  6,  4,  2,  0,  8,  6,  4;
   9,  8,  7,  6,  5,  4,  3,  2,  1;
  10, 20, 30, 40, 50, 60, 70, 80, 90, 100;
  11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121;
  12, 24, 36, 48, 50, 62, 74, 86, 98, 120, 132, 144;

Seiichi Manyama, Rows n = 1..99, flattened
David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.
Index entries for sequences related to carryless arithmetic

T(n,n) gives A059729.

Cf. A001477 for carryless 1 X n, A004520 for carryless 2 X n base 10, A055120 for carryless 9 X n, A008592 for carryless 10 X n, A059691 for carryless 12 X n.

T(n, k) = fromdigits(Vec(Pol(digits(n))*Pol(digits(k)))%10);

A170986 Carryless product n X n in base 5.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Aug 29 2010

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

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

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

A170987 Carryless product n X n in base 6.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Aug 29 2010

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

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

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

A170988 Carryless product n X n in base 7.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Aug 29 2010

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

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

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

A170989 Carryless product n X n in base 8.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Aug 29 2010

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

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

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

A361390 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) is carryless n^k base 10.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 1, 0, 1, 8, 9, 4, 1, 0, 1, 6, 7, 6, 5, 1, 0, 1, 2, 1, 4, 5, 6, 1, 0, 1, 4, 3, 6, 5, 6, 7, 1, 0, 1, 8, 9, 4, 5, 6, 9, 8, 1, 0, 1, 6, 7, 6, 5, 6, 3, 4, 9, 1, 0, 1, 2, 1, 4, 5, 6, 1, 2, 1, 10, 1, 0, 1, 4, 3, 6, 5, 6, 7, 6, 9, 100, 11, 1, 0, 1, 8, 9, 4, 5, 6, 9, 8, 1, 1000, 121, 12, 1

Offset: 0

Seiichi Manyama, Mar 10 2023

			4 * 4 = 16, so T(4,2) = 6. 6 * 4 = 24, so T(4,3) = 4.
Square array begins:
  1, 0, 0, 0, 0, 0, 0, 0, ...
  1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 2, 4, 8, 6, 2, 4, 8, ...
  1, 3, 9, 7, 1, 3, 9, 7, ...
  1, 4, 6, 4, 6, 4, 6, 4, ...
  1, 5, 5, 5, 5, 5, 5, 5, ...
  1, 6, 6, 6, 6, 6, 6, 6, ...
  1, 7, 9, 3, 1, 7, 9, 3, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.
Index entries for sequences related to carryless arithmetic

Columns k=0..4 give A000012, A001477, A059729, A169885, A169886.
Rows n=0..4 give A000007, A000012, A000689, A001148, A168428.
T(11,k) gives A059734.
Main diagonal gives A361351.
Cf. A004248, A359697.

T(n, k) = fromdigits(Vec(Pol(digits(n))^k)%10);

cp's OEIS Frontend

A359697 Triangle T(n,k), n >= 1, 1 <= k <= n, read by rows, where T(n,k) is carryless product n X k base 10.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A170986 Carryless product n X n in base 5.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A170987 Carryless product n X n in base 6.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A170988 Carryless product n X n in base 7.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A170989 Carryless product n X n in base 8.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A361390 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) is carryless n^k base 10.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI