A004520 -id:A004520

A059729

Carryless squares n X n base 10.

+10
27

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

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

OFFSET

0,3

LINKS

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

EXAMPLE

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

PROG

(Python)

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

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

CROSSREFS

Cf. A004520, A059692, A169889, A169963.

See A087019 (lunar squares) for another version.

KEYWORD

base,easy,nonn,look

AUTHOR

Henry Bottomley, Feb 20 2001

STATUS

approved

A169887

Primes in carryless arithmetic mod 10.

+10
9

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

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

OFFSET

1,1

COMMENTS

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).

LINKS

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

EXAMPLE

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.

CROSSREFS

A169887 = A163396 union A169984.

Cf. A004520, A059729, A168294, A168541, A169885, A169886, A169884, A169903 (primitive primes).

Cf. A169962.

KEYWORD

nonn,base

AUTHOR

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

STATUS

approved

A059692

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

+10
7

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

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

OFFSET

0,8

LINKS

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

EXAMPLE

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.

MATHEMATICA

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 *)

PROG

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

CROSSREFS

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).

KEYWORD

nonn,base,easy,tabl,look

AUTHOR

Henry Bottomley, Feb 19 2001

EXTENSIONS

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

STATUS

approved

A168294

Carryless product n times n+1.

+10
7

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

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

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..58.

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

(Python)

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

CROSSREFS

Cf. A004520, A059729.

KEYWORD

nonn,base

AUTHOR

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

STATUS

approved

A169885

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

+10
7

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

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

OFFSET

0,3

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..9999

Index entries for sequences related to carryless arithmetic

PROG

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

CROSSREFS

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

KEYWORD

nonn,base,look

AUTHOR

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

STATUS

approved

A169886

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

+10
7

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

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

OFFSET

0,3

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..9999

Index entries for sequences related to carryless arithmetic

PROG

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

CROSSREFS

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

KEYWORD

nonn,base,look

AUTHOR

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

STATUS

approved

A169884

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

+10
6

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

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

OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..57.

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

MATHEMATICA

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 *)

CROSSREFS

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

KEYWORD

nonn,base,easy

AUTHOR

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

STATUS

approved

A168541

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

+10
5

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

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

OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..41.

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

MATHEMATICA

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 *)

CROSSREFS

Cf. A004520, A059729, A168294, A169884.

KEYWORD

nonn,base,easy

AUTHOR

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

EXTENSIONS

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

STATUS

approved

A169894

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

+10
4

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

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

OFFSET

0,4

LINKS

Stefano Spezia, First 151 antidiagonals of the table, flattened

Index entries for sequences related to carryless arithmetic

EXAMPLE

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 ...

...

MAPLE

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

MATHEMATICA

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 *)

CROSSREFS

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

KEYWORD

nonn,base,look,tabl

AUTHOR

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

STATUS

approved

A059632

Carryless product 11 X n base 10.

+10
3

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

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

OFFSET

0,2

COMMENTS

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

LINKS

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.

EXAMPLE

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

PROG

(Haskell)

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

CROSSREFS

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.

KEYWORD

base,nonn

AUTHOR

Henry Bottomley, Feb 19 2001

STATUS

approved