A004520 -id:A004520 - cp's OEIS frontend

A059691 Carryless product 12 X n base 10.

0, 12, 24, 36, 48, 50, 62, 74, 86, 98, 120, 132, 144, 156, 168, 170, 182, 194, 106, 118, 240, 252, 264, 276, 288, 290, 202, 214, 226, 238, 360, 372, 384, 396, 308, 310, 322, 334, 346, 358, 480, 492, 404, 416, 428, 430, 442, 454, 466, 478, 500, 512, 524, 536

Offset: 0

Henry Bottomley, Feb 19 2001

			a(97)=954 since we have 12 X 97 = carryless sum of 900, 80, 70 and 4 = 954

Seiichi Manyama, Table of n, a(n) for n = 0..9999
David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.
Index entries for sequences related to carryless arithmetic

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.

A059734 Carryless 11^n base 10; a(n) is carryless sum of 10*a(n-1) and a(n-1).

Original entry on oeis.org

1, 11, 121, 1331, 14641, 150051, 1650561, 17155171, 188606881, 1964664691, 10500200501, 115502205511, 1260524250621, 13865766756831, 141412323214141, 1555535555355551, 16000880008800061, 176008680086800671

Offset: 0

Henry Bottomley, Feb 20 2001

Subsequence of A002113. - Chai Wah Wu, Jul 30 2025

			a(7)=17155171 since a(6)=1650561 and digits of a(7) are sum mod 10 of 1, 6+1=7, 5+6=1, 0+5=5, 5+0=5, 6+5=1, 1+6=7 and 1.

Seiichi Manyama, Table of n, a(n) for n = 0..999
David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version

Cf. A004520, A006940, A008975, A361351, A002113.

Table[Sum[Mod[Binomial[n, m], 10]*10^m, {m, 0, n}], {n, 0, 30}] (* Roger L. Bagula and Gary W. Adamson, Sep 14 2008 *)

a(n) = fromdigits(Vec(Pol(digits(11))^n)%10); \\ Seiichi Manyama, Mar 10 2023

from math import comb, prod
from sympy.ntheory.modular import crt
from gmpy2 import digits
def A059734(n):
    k, l = 0, len(s:=digits(n,5))
    for m in range(n+1):
        t = digits(m,5).zfill(l)
        k = 10*k+crt([5,2],[prod(comb(int(s[i]),int(t[i]))%5 for i in range(l))%5,int(not ~n & m)])[0]
    return k # Chai Wah Wu, Jul 30 2025

a(n)=Sum[Mod[Binomial[n, m], 10]*10^m, {m, 0, n}]. - Roger L. Bagula and Gary W. Adamson, Sep 14 2008

A169903 Primitive primes in carryless arithmetic mod 10.

Original entry on oeis.org

21, 23, 25, 27, 29, 51, 56, 201, 209, 227, 229, 241, 243, 261, 263, 287, 289, 551, 2023, 2027, 2043, 2047, 2061, 2069, 2081, 2089, 2207, 2209, 2221, 2223, 2263, 2267, 2281, 2287, 2401, 2407, 2421, 2423, 2441, 2449, 2483, 2489, 2603, 2609

Offset: 1

David Applegate, Marc LeBrun and N. J. A. Sloane, Jul 11 2010

Define the units in carryless arithmetic mod 10 to be the numbers 1, 3, 7 and 9 (these divide any number). A prime is a number N, not a unit, whose only factorizations are of the form N = u * M, where u is a unit.
A prime is primitive if it is not the carryless product of a smaller prime and a unit.
A subsequence of A169887.

David Applegate, Table of n, a(n) for n = 1..843 (all primitive primes with <= 6 digits)
Index entries for sequences related to carryless arithmetic

Cf. A004520, A059729, A168294, A168541, A169885, A169886, A169884, A169887.

Cf. A058943, A058945.

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.

Original entry on oeis.org

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);

A059626 Generalized nim sum n + n + n in base 10; carryless multiplication 3 X n base 10.

Original entry on oeis.org

0, 3, 6, 9, 2, 5, 8, 1, 4, 7, 30, 33, 36, 39, 32, 35, 38, 31, 34, 37, 60, 63, 66, 69, 62, 65, 68, 61, 64, 67, 90, 93, 96, 99, 92, 95, 98, 91, 94, 97, 20, 23, 26, 29, 22, 25, 28, 21, 24, 27, 50, 53, 56, 59, 52, 55, 58, 51, 54, 57, 80, 83, 86, 89, 82, 85, 88, 81, 84, 87, 10, 13

Offset: 0

Henry Bottomley, Feb 19 2001

nonn

David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.
Index entries for sequences that are permutations of the natural numbers

Cf. A001477 for carryless 1 X n, A004520 for carryless 2 X 10 base 10, A008592 for carryless 10 X n.

A059629 Carryless multiplication 6 X n base 10.

Original entry on oeis.org

0, 6, 2, 8, 4, 0, 6, 2, 8, 4, 60, 66, 62, 68, 64, 60, 66, 62, 68, 64, 20, 26, 22, 28, 24, 20, 26, 22, 28, 24, 80, 86, 82, 88, 84, 80, 86, 82, 88, 84, 40, 46, 42, 48, 44, 40, 46, 42, 48, 44, 0, 6, 2, 8, 4, 0, 6, 2, 8, 4, 60, 66, 62, 68, 64, 60, 66, 62, 68, 64, 20, 26, 22, 28, 24

Offset: 0

Henry Bottomley, Feb 19 2001

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

Cf. A001477 for carryless 1 X n, A004520 for carryless 2 X 10 base 10, A008592 for carryless 10 X n.

A059631 Carryless multiplication 8 X n base 10.

Original entry on oeis.org

0, 8, 6, 4, 2, 0, 8, 6, 4, 2, 80, 88, 86, 84, 82, 80, 88, 86, 84, 82, 60, 68, 66, 64, 62, 60, 68, 66, 64, 62, 40, 48, 46, 44, 42, 40, 48, 46, 44, 42, 20, 28, 26, 24, 22, 20, 28, 26, 24, 22, 0, 8, 6, 4, 2, 0, 8, 6, 4, 2, 80, 88, 86, 84, 82, 80, 88, 86, 84, 82, 60, 68, 66, 64, 62

Offset: 0

Henry Bottomley, Feb 19 2001

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

Cf. A001477 for carryless 1 X n, A004520 for carryless 2 X 10 base 10, A008592 for carryless 10 X n.

A059627 Generalized nim sum n + n + n + n in base 10; carryless multiplication 4 X n base 10.

Original entry on oeis.org

0, 4, 8, 2, 6, 0, 4, 8, 2, 6, 40, 44, 48, 42, 46, 40, 44, 48, 42, 46, 80, 84, 88, 82, 86, 80, 84, 88, 82, 86, 20, 24, 28, 22, 26, 20, 24, 28, 22, 26, 60, 64, 68, 62, 66, 60, 64, 68, 62, 66, 0, 4, 8, 2, 6, 0, 4, 8, 2, 6, 40, 44, 48, 42, 46, 40, 44, 48, 42, 46, 80, 84, 88, 82, 86

Offset: 0

Henry Bottomley, Feb 19 2001

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

Cf. A001477 for carryless 1 X n, A004520 for carryless 2 X 10 base 10, A008592 for carryless 10 X n.

A059628 Carryless multiplication 5 X n base 10.

Original entry on oeis.org

0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 50, 55, 50, 55, 50, 55, 50, 55, 50, 55, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 50, 55, 50, 55, 50, 55, 50, 55, 50, 55, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 50, 55, 50, 55, 50, 55, 50, 55, 50, 55, 0, 5, 0, 5, 0, 5, 0, 5, 0, 5, 50, 55, 50, 55, 50, 55, 50, 55, 50, 55, 0, 5, 0, 5

Offset: 0

Henry Bottomley, Feb 19 2001

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

Cf. A001477 for carryless 1 X n, A004520 for carryless 2 X 10 base 10, A008592 for carryless 10 X n.

A059630 Carryless multiplication 7 X n base 10.

Original entry on oeis.org

0, 7, 4, 1, 8, 5, 2, 9, 6, 3, 70, 77, 74, 71, 78, 75, 72, 79, 76, 73, 40, 47, 44, 41, 48, 45, 42, 49, 46, 43, 10, 17, 14, 11, 18, 15, 12, 19, 16, 13, 80, 87, 84, 81, 88, 85, 82, 89, 86, 83, 50, 57, 54, 51, 58, 55, 52, 59, 56, 53, 20, 27, 24, 21, 28, 25, 22, 29, 26, 23, 90, 97

Offset: 0

Henry Bottomley, Feb 19 2001

David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.
Index entries for sequences that are permutations of the natural numbers

Cf. A001477 for carryless 1 X n, A004520 for carryless 2 X 10 base 10, A008592 for carryless 10 X n.

cp's OEIS Frontend

A059691 Carryless product 12 X n base 10.

Views

Author

Keywords

Examples

Links

Crossrefs

A059734 Carryless 11^n base 10; a(n) is carryless sum of 10*a(n-1) and a(n-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A169903 Primitive primes in carryless arithmetic mod 10.

Views

Author

Keywords

Comments

Links

Crossrefs

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

A059626 Generalized nim sum n + n + n in base 10; carryless multiplication 3 X n base 10.

Views

Author

Keywords

Links

Crossrefs

A059629 Carryless multiplication 6 X n base 10.

Views

Author

Keywords

Links

Crossrefs

A059631 Carryless multiplication 8 X n base 10.

Views

Author

Keywords

Links

Crossrefs

A059627 Generalized nim sum n + n + n + n in base 10; carryless multiplication 4 X n base 10.

Views

Author

Keywords

Links

Crossrefs

A059628 Carryless multiplication 5 X n base 10.

Views

Author

Keywords

Links

Crossrefs

A059630 Carryless multiplication 7 X n base 10.

Views

Author

Keywords

Links

Crossrefs