A004520 -id:A004520 - cp's OEIS frontend

A059729 Carryless squares n X n base 10.

0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 100, 121, 144, 169, 186, 105, 126, 149, 164, 181, 400, 441, 484, 429, 466, 405, 446, 489, 424, 461, 900, 961, 924, 989, 946, 905, 966, 929, 984, 941, 600, 681, 664, 649, 626, 605, 686, 669, 644, 621, 500, 501, 504, 509, 506, 505

Offset: 0

Henry Bottomley, Feb 20 2001

			a(87) is carryless sum of (6)400, (5)60, (5)60 and (4)9, i.e., 400+20+9 = 429.

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. A004520, A059692, A169889, A169963.

See A087019 (lunar squares) for another version.

a(n) = fromdigits(Vec(Pol(digits(n))^2)%10) \\ Ruud H.G. van Tol, Dec 07 2022

def A059729(n):
    s = [int(d) for d in str(n)]
    l = len(s)
    t = [0]*(2*l-1)
    for i in range(l):
        for j in range(l):
            t[i+j] = (t[i+j] + s[i]*s[j]) % 10
    return int("".join(str(d) for d in t)) # Chai Wah Wu, Jun 29 2020

A169887 Primes in carryless arithmetic mod 10.

Original entry on oeis.org

21, 23, 25, 27, 29, 41, 43, 45, 47, 49, 51, 52, 53, 54, 56, 57, 58, 59, 61, 63, 65, 67, 69, 81, 83, 85, 87, 89, 201, 209, 227, 229, 241, 243, 261, 263, 287, 289, 403, 407, 421, 427, 443, 449, 463, 469, 481, 487, 551, 553, 557, 559, 603, 607, 623, 629, 641, 647, 661, 667, 683, 689, 801, 809, 821, 823, 847, 849, 867, 869, 881, 883

Offset: 1

David Applegate, Marc LeBrun and N. J. A. Sloane, Jul 06 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.

There are two types: e-type primes (A163396) and f-type (A169984).

			Examples of nonprimes: 2 = 2*51, 4 = 2*2, 10 = 52*85, 11 = 57*83, 101 = 13*17, 102 = 58 * 254 = 502 * 801, 103 = 53 * 251 = 507 * 809, 107 = 53 * 259 = 507 * 801, 108 = 58 * 256 = 502 * 809, 111 = 227 * 553.

N. J. A. Sloane, Table of n, a(n) for n = 1..253560 (All primes with <= 9 digits. Based on T. D. Noe's b-files for A058943 and A058955.)
David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.
Index entries for sequences related to carryless arithmetic

A169887 = A163396 union A169984.
Cf. A004520, A059729, A168294, A168541, A169885, A169886, A169884, A169903 (primitive primes).
Cf. A169962.

A059692 Table of carryless products i * j, i>=0, j>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 4, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 8, 9, 8, 5, 0, 0, 6, 0, 2, 2, 0, 6, 0, 0, 7, 2, 5, 6, 5, 2, 7, 0, 0, 8, 4, 8, 0, 0, 8, 4, 8, 0, 0, 9, 6, 1, 4, 5, 4, 1, 6, 9, 0, 0, 10, 8, 4, 8, 0, 0, 8, 4, 8, 10, 0, 0, 11, 20, 7, 2, 5, 6, 5, 2, 7, 20, 11, 0

Offset: 0

Henry Bottomley, Feb 19 2001

			Table begins:
  0, 0, 0, 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, 13, 14, 15 ...
  0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 20, 22, 24, 26, 28, 20 ...
  0, 3, 6, 9, 2, 5, 8, 1, 4, 7, 30, 33, 36, 39, 32, 35 ...
  0, 4, 8, 2, 6, 0, 4, 8, 2, 6, 40, 44, 48, 42, 46, 40 ...
  ...
T(12, 97) = 954 since we have 12 X 97 = carryless sum of 900, (180 mod 100=)80, 70 and (14 mod 10=)4 = 954.

Stefano Spezia, First 140 antidiagonals of the table, 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

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

Cf. A048720 (binary), A325820 (ternary).

len[num_]:=Length[IntegerDigits[num]]; digit[num_,d_]:=Part[IntegerDigits[num],d]; T[i_, j_] := FromDigits[Reverse[CoefficientList[PolynomialMod[Sum[digit[i,c]*x^(len[i]-c), {c, len[i]}]*Sum[digit[j,r]*x^(len[j]-r), {r, len[j]}], 10], x]]]; Flatten[Table[T[i - j, j], {i, 0, 12}, {j, 0, i}]] (* Stefano Spezia, Sep 26 2022 *)

T(n,k) = fromdigits(lift(Vec( Mod(Pol(digits(n)),10) * Pol(digits(k))))); \\ Kevin Ryde, Sep 27 2022

Minor edits by N. J. A. Sloane, Aug 24 2010

A168294 Carryless product n times n+1.

Original entry on oeis.org

0, 2, 6, 2, 0, 0, 2, 6, 2, 90, 110, 132, 156, 172, 190, 110, 132, 156, 172, 280, 420, 462, 406, 442, 480, 420, 462, 406, 442, 670, 930, 992, 956, 912, 970, 930, 992, 956, 912, 260, 640, 622, 606, 682, 660, 640, 622, 606, 682, 50, 550, 552, 556, 552, 550, 550, 552, 556, 552

Offset: 0

David Applegate, Marc LeBrun and N. J. A. Sloane, Jul 06 2010

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. A004520, A059729.

def A168294(n):
    s, t = [int(d) for d in str(n)], [int(d) for d in str(n+1)]
    l, m = len(s), len(t)
    u = [0]*(l+m-1)
    for i in range(l):
        for j in range(m):
            u[i+j] = (u[i+j] + s[i]*t[j]) % 10
    return int("".join(str(d) for d in u)) # Chai Wah Wu, Jun 30 2020

A169885 Cubes (n * n * n) in carryless arithmetic mod 10.

Original entry on oeis.org

0, 1, 8, 7, 4, 5, 6, 3, 2, 9, 1000, 1331, 1628, 1977, 1284, 1555, 1886, 1173, 1422, 1739, 8000, 8261, 8448, 8647, 8864, 8005, 8266, 8443, 8642, 8869, 7000, 7791, 7468, 7117, 7844, 7555, 7246, 7913, 7662, 7399, 4000, 4821, 4688, 4487, 4224, 4005, 4826, 4683

Offset: 0

David Applegate, Marc LeBrun and N. J. A. Sloane, Jul 06 2010

Seiichi Manyama, Table of n, a(n) for n = 0..9999
Index entries for sequences related to carryless arithmetic

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

a(n) = fromdigits(Vec(Pol(digits(n))^3)%10); \\ Seiichi Manyama, Mar 09 2023

A169886 Fourth powers (n * n * n * n) in carryless arithmetic mod 10.

Original entry on oeis.org

0, 1, 6, 1, 6, 5, 6, 1, 6, 1, 10000, 14641, 18426, 12481, 16666, 10005, 14646, 18421, 12486, 16661, 60000, 62481, 64646, 66661, 68426, 60005, 62486, 64641, 66666, 68421, 10000, 18421, 16666, 14641, 12486, 10005, 18426, 16661, 14646, 12481, 60000, 66661, 62486

Offset: 0

David Applegate, Marc LeBrun and N. J. A. Sloane, Jul 06 2010

Seiichi Manyama, Table of n, a(n) for n = 0..9999
Index entries for sequences related to carryless arithmetic

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

a(n) = fromdigits(Vec(Pol(digits(n))^4)%10); \\ Seiichi Manyama, Mar 09 2023

A169884 Numbers consisting of either all even digits or just 5's and 0's.

Original entry on oeis.org

0, 2, 4, 5, 6, 8, 20, 22, 24, 26, 28, 40, 42, 44, 46, 48, 50, 55, 60, 62, 64, 66, 68, 80, 82, 84, 86, 88, 200, 202, 204, 206, 208, 220, 222, 224, 226, 228, 240, 242, 244, 246, 248, 260, 262, 264, 266, 268, 280, 282, 284, 286, 288, 400, 402, 404, 406

Offset: 1

David Applegate, Marc LeBrun and N. J. A. Sloane, Jul 06 2010

These are all the divisors of zero in carryless arithmetic mod 10. E.g. 5 * 44 = 0.

David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.
Index entries for 10-automatic sequences.
Index entries for sequences related to carryless arithmetic

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

With[{upto=410},Select[Union[Join[Select[Range[upto],And@@EvenQ[ IntegerDigits[#]]&], FromDigits/@Tuples[{5,0},Ceiling[Log[ 10,upto]]]]],#<=upto&]] (* Harvey P. Dale, Aug 05 2011 *)
elect[Range[0,500],AllTrue[IntegerDigits[#],EvenQ]||SubsetQ[{0,5},IntegerDigits[#]]&] (* Harvey P. Dale, Aug 22 2025 *)

A168541 Numbers consisting of either 2's and 0's or 5's and 0's.

Original entry on oeis.org

2, 5, 20, 50, 200, 202, 220, 500, 505, 550, 2000, 2002, 2020, 2022, 2200, 2202, 2220, 5000, 5005, 5050, 5055, 5500, 5505, 5550, 20000, 20002, 20020, 20022, 20200, 20202, 20220, 20222, 22000, 22002, 22020, 22022, 22200, 22202, 22220, 50000, 50005

Offset: 1

David Applegate, Marc LeBrun and N. J. A. Sloane, Jul 06 2010

A subset of the divisors of zero in carryless arithmetic mod 10, e.g., 5 * 44 = 0.

David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.
Index entries for 10-automatic sequences.
Index entries for sequences related to carryless arithmetic

Cf. A004520, A059729, A168294, A169884.

lst = {2, 5}; k = 1; While[k < 10^5, id = Union@ IntegerDigits@k; len = Length@ id; If[ len == 2 && id == {0, 2} || id == {0, 5}, AppendTo[lst, k]]; k++ ]; lst (* Robert G. Wilson v, Jul 12 2010 *)
Join[{2,5},Sort[Flatten[Table[Select[FromDigits/@Tuples[{k,0},6],DigitCount[ #,10,0]>0 && DigitCount[#,10,k]>0&],{k,{2,5}}]]]] (* Harvey P. Dale, Jul 03 2020 *)

More terms from Robert G. Wilson v, Jul 12 2010

A169894 Table of carryless sums i + j, i>=0, j>=0, read by antidiagonals.

Original entry on oeis.org

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

Offset: 0

David Applegate, Marc LeBrun and N. J. A. Sloane, Jul 06 2010

			Table begins:
  0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 ...
  1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 11, 12, 13, 14, 15, 16 ...
  2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 12, 13, 14, 15, 16, 17 ...
  3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 13, 14, 15, 16, 17, 18 ...
  4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 14, 15, 16, 17, 18, 19 ...
  ...

Cf. A004520 (diagonal), A059692 (carryless products).

A169894 := proc(a,b)
    local adigs,bdigs,cdigs ;
    adigs := convert(a,base,10) ;
    bdigs := convert(b,base,10) ;
    len := max(nops(adigs),nops(bdigs)) ;
    adigs := [op(adigs),seq(0,d=1..len-nops(adigs))] ;
    bdigs := [op(bdigs),seq(0,d=1..len-nops(bdigs))] ;
    cdigs := [] ;
    for d from 1 to len do
        cdigs := [op(cdigs),A010879(op(d,adigs)+op(d,bdigs))] ;
    end do:
    add(op(d,cdigs)*10^(d-1),d=1..len) ;
end proc: # R. J. Mathar, Jul 12 2013

len[num_]:=Length[IntegerDigits[num]]; digit[num_, d_]:=Part[IntegerDigits[num], d]; T[i_, j_] := FromDigits[Reverse[CoefficientList[PolynomialMod[Sum[digit[i, c]*x^(len[i]-c), {c, len[i]}]+Sum[digit[j, r]*x^(len[j]-r), {r, len[j]}], 10], x]]]; Table[T[i - j, j], {i, 0, 12}, {j, 0, i}] (* Stefano Spezia, Dec 20 2023 *)

A059632 Carryless product 11 X n base 10.

Original entry on oeis.org

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 143, 154, 165, 176, 187, 198, 109, 220, 231, 242, 253, 264, 275, 286, 297, 208, 219, 330, 341, 352, 363, 374, 385, 396, 307, 318, 329, 440, 451, 462, 473, 484, 495, 406, 417, 428, 439, 550, 561, 572, 583

Offset: 0

Henry Bottomley, Feb 19 2001

a(n) <= 11*n; a(m) = 11*m iff m is a term of A039691. - Reinhard Zumkeller, Jul 05 2014

			a(19)=109 since we have 11 X 19 = carryless sum of 100, 90, 10 and 9 =109

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
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, A055120 for carryless 9 X n, A008592 for carryless 10 X n.
Cf. A048724 carryless 3Xn in base 2, A242399 carryless 4Xn in base 3.
Cf. A008593.

a059632 n = foldl (\v d -> 10 * v + d) 0 $
                  map (flip mod 10) $ zipWith (+) ([0] ++ ds) (ds ++ [0])
            where ds = map (read . return) $ show n
-- Reinhard Zumkeller, Jul 05 2014

cp's OEIS Frontend

A059729 Carryless squares n X n base 10.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Python

A169887 Primes in carryless arithmetic mod 10.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A059692 Table of carryless products i * j, i>=0, j>=0, read by antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A168294 Carryless product n times n+1.

Views

Author

Keywords

Links

Crossrefs

Programs

Python

A169885 Cubes (n * n * n) in carryless arithmetic mod 10.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A169886 Fourth powers (n * n * n * n) in carryless arithmetic mod 10.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A169884 Numbers consisting of either all even digits or just 5's and 0's.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A168541 Numbers consisting of either 2's and 0's or 5's and 0's.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A169894 Table of carryless sums i + j, i>=0, j>=0, read by antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs