A059729 -id:A059729 - cp's OEIS frontend

A063010 Carryless binary square of n; also Moser-de Bruijn sequence written in binary.

0, 1, 100, 101, 10000, 10001, 10100, 10101, 1000000, 1000001, 1000100, 1000101, 1010000, 1010001, 1010100, 1010101, 100000000, 100000001, 100000100, 100000101, 100010000, 100010001, 100010100, 100010101, 101000000, 101000001

Offset: 0

Henry Bottomley, Jul 03 2001

Numbers that are sums of distinct powers of 100. - David Wasserman, Feb 26 2008

			a(11)=1000101, since 11 in binary is 1011 and binary carryless sum of 1011000, 0, 10110 and 1011 is 1000101.

Michael De Vlieger, Table of n, a(n) for n = 0..8191
David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 45.

Cf. Moser-de Bruijn sequence A000695, carryless decimal squares A059729, pre-carry binary squares A063009.

With[{k = 100}, Map[FromDigits[#, k] &, Tuples[{0, 1}, 5]]] (* Michael De Vlieger, Oct 29 2022 *)

a(n) = fromdigits(binary(n),100); \\ Ruud H.G. van Tol, Dec 05 2022

def A063010(n): return int(bin(int(bin(n)[2:],4))[2:]) # Chai Wah Wu, Apr 09 2025

a(n) = A062033(n)/10, i.e., with final zero removed.
a(n) = Sum_{k>=0} A030308(n,k)*A098608(k). - Philippe Deléham, Oct 15 2011
G.f.: (1/(1 - x))*Sum_{k>=0} 100^k*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jun 04 2017

More terms from David Wasserman, Feb 26 2008

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

A169890 Carryless sum 1+2+3+...+n.

Original entry on oeis.org

0, 1, 3, 6, 0, 5, 1, 8, 6, 5, 15, 26, 38, 41, 55, 60, 76, 83, 91, 0, 20, 41, 63, 86, 0, 25, 41, 68, 86, 5, 35, 66, 98, 21, 55, 80, 16, 43, 71, 0, 40, 81, 23, 66, 0, 45, 81, 28, 66, 5, 55, 6, 58, 1, 55, 0, 56, 3, 51, 0, 60, 21, 83, 46, 0, 65, 21, 88, 46, 5, 75, 46, 18, 81, 55, 20, 96, 63, 31

Offset: 0

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

Carryless analog of triangular numbers.

Rémy Sigrist, 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. A004250, A059729.

A339023 Replace each digit d in the decimal representation of n with the digital root of n*d.

Original entry on oeis.org

0, 1, 4, 9, 7, 7, 9, 4, 1, 9, 10, 22, 36, 43, 52, 63, 76, 82, 99, 19, 40, 63, 88, 16, 36, 58, 73, 99, 28, 49, 90, 34, 61, 99, 31, 64, 99, 37, 67, 99, 70, 25, 63, 13, 55, 99, 46, 85, 36, 79, 70, 36, 85, 46, 99, 55, 13, 63, 25, 79, 90, 67, 37, 99, 64, 31, 99, 61

Offset: 0

Sebastian Karlsson, Jan 18 2021

			a(23) = 16 because 2*23 = 46 and 3*23 = 69 and the digital roots of 46 and 69 are 1 and 6.

Sebastian Karlsson, Table of n, a(n) for n = 0..10000
Sebastian Karlsson, The sequence generalized and plotted in arbitrary bases

Cf. A010888, A056992, A059729, A106649, A106746, A139337, A316347, A322131.

dr(n) = if(n, (n-1)%9+1); \\ A010888
a(n) = if (n==0, return(0)); my(d=digits(n), s=""); for (k=1, #d, s=concat(s, dr(n*d[k]))); eval(s); \\ Michel Marcus, Jan 18 2021

def digitalroot(n):
    return 0 if n == 0 else (n-1)%9 + 1
def a(n):
    return int(''.join([str(digitalroot(n*int(d))) for d in str(n)]))
for n in range(0, 68):
    print(a(n), end=', ')

a(9*n + 1) = 9*n + 1.

a(10*n) = 10*a(n). - Sebastian Karlsson, Feb 14 2021

A169916 Squares in carryless arithmetic mod 10 with addition and multiplication of digits both defined to be addition mod 10.

Original entry on oeis.org

0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 220, 242, 264, 286, 208, 220, 242, 264, 286, 208, 440, 462, 484, 406, 428, 440, 462, 484, 406, 428, 660, 682, 604, 626, 648, 660, 682, 604, 626, 648, 880, 802, 824, 846, 868, 880, 802, 824, 846, 868, 0, 22, 44, 66, 88, 0, 22, 44, 66, 88, 220, 242

Offset: 0

David Applegate, Marc LeBrun and N. J. A. Sloane, Jul 20 2010

The rules of arithmetic used in A169916, A169917, A169918 have very strange consequences. Many of the familiar laws fail. For instance, the arithmetic in A169916 is not associative: 10*(9*2) = 10*1 = 21 != (10*9)*2 = 9*2 = 1.

			a(16) = 16*16 = 242:
....16
....16
------
....72 (6*6 = 6+6 mod 10 = 2, 6*1 = 6+1 mod 10 = 7)
...27.
------
...242
------

Index entries for sequences related to carryless arithmetic

The four versions are A059729, A169916, A169917, A169918.

A169916(n)={u=vector(#n=digits(n),i,1);n=apply(d->n+d*u,n)%10;sum(i=0,2*#n-2,sum(j=max(1,#n-i),min(2*#n-1-i,#n),n[2*#n-i-j][j])%10*10^i)} \\ M. F. Hasler, Mar 26 2015

a(n)=a(n') if respective digits of n and n' differ by 0 or 5. In particular, a(10k+m) = a(10k+m+5) if 0 <= m <= 4.

A169917 Squares in carryless arithmetic mod 10 with addition and multiplication of digits both defined to be multiplication mod 10.

Original entry on oeis.org

0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 100, 111, 144, 199, 166, 155, 166, 199, 144, 111, 400, 441, 464, 469, 446, 405, 446, 469, 464, 441, 900, 991, 964, 919, 946, 955, 946, 919, 964, 991, 600, 661, 644, 649, 666, 605, 666, 649, 644, 661, 500, 551, 504, 559, 506, 555, 506, 559, 504

Offset: 0

David Applegate, Marc LeBrun and N. J. A. Sloane, Jul 20 2010

The rules of arithmetic used in A169916, A169917, A169918 have very strange consequences. Many of the familiar laws fail. For instance, the arithmetic in A169916 is not associative: 10*(9*2) = 10*1 = 21 != (10*9)*2 = 9*2 = 1.

			a(24) = 24*24 = 446:
...24
...24
-----
...86
..48.
-----
..446
(The rule for "adding" the columns is to multiply mod 10: 8+8 = 8 * 8 mod 10 = 4.)

Index entries for sequences related to carryless arithmetic

The four versions are A059729, A169916, A169917, A169918.

A169917(n)={#n=digits(n);n=apply(d->n*d,n)%10;sum(i=0,2*#n-2,prod(j=max(1,#n-i),min(2*#n-1-i,#n),n[2*#n-i-j][j])%10*10^i)} \\ M. F. Hasler, Mar 26 2015

a(n) = a(n') if the i-th digit of n' either equals the i-th digit of n or (10 - the i-th digit of n): e.g., a(12345) = a(18365), because the 2nd and 4th digit of 12345 equal 10-(the 2nd resp. 4th digit of 18365), and the other digits are the same. In particular, a(10k+5+m) = a(10k+5-m), for m=0,...,4. - M. F. Hasler, Mar 26 2015

A169963 Number of (2n+1)-digit squares in carryless arithmetic mod 10.

Original entry on oeis.org

5, 46, 452, 4504, 45008, 450016, 4500032, 45000064, 450000128, 4500000256, 45000000512, 450000001024, 4500000002048, 45000000004096, 450000000008192, 4500000000016384, 45000000000032768, 450000000000065536, 4500000000000131072, 45000000000000262144

Offset: 0

N. J. A. Sloane, Aug 07 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
Index entries for linear recurrences with constant coefficients, signature (12,-20).

See A059729 for the actual squares.

f :- n->2^((n-1)/2) + add( 5^d*2^((n+1)/2),d=0..(n-3)/2) + 2^((n+3)/2)*5^((n-1)/2);

LinearRecurrence[{12, -20}, {5, 46}, 25] (* Paolo Xausa, Jun 26 2024 *)

For formula see Maple code.

a(n) = 12*a(n-1)-20*a(n-2). G.f.: -(14*x-5) / ((2*x-1)*(10*x-1)). - Colin Barker, May 11 2013

A361351 Carryless n-th powers of n base 10.

Original entry on oeis.org

1, 1, 4, 7, 6, 5, 6, 3, 6, 9, 10000000000, 115502205511, 1440046600466, 19225142754633, 166668888866666, 1555555555555555, 16000880008800066, 194006440028800877, 1422046880284402844, 11116222228888849999, 600000000000000000000, 2600042000840006800021

Offset: 0

Seiichi Manyama, Mar 09 2023

Seiichi Manyama, Table of n, a(n) for n = 0..499
Index entries for sequences related to carryless arithmetic

Cf. A059729, A169885, A169886.

a(n) = fromdigits(Vec(Pol(digits(n))^n)%10);

A169889 Numbers that are carryless squares in base 10.

Original entry on oeis.org

0, 1, 4, 5, 6, 9, 100, 105, 121, 126, 144, 149, 164, 169, 181, 186, 400, 405, 424, 429, 441, 446, 461, 466, 484, 489, 500, 501, 504, 505, 506, 509, 600, 605, 621, 626, 644, 649, 664, 669, 681, 686, 900, 905, 924, 929, 941, 946, 961, 966, 984, 989, 10000, 10005, 10104

Offset: 1

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

A059729 sorted and duplicates removed.

N. J. A. Sloane, Table of n, a(n) for n = 1..5008
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. A059729, A169963.

See A087019 (lunar squares) for another version.

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.

cp's OEIS Frontend

A063010 Carryless binary square of n; also Moser-de Bruijn sequence written in binary.

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

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A169890 Carryless sum 1+2+3+...+n.

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A339023 Replace each digit d in the decimal representation of n with the digital root of n*d.

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A169916 Squares in carryless arithmetic mod 10 with addition and multiplication of digits both defined to be addition mod 10.

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A169917 Squares in carryless arithmetic mod 10 with addition and multiplication of digits both defined to be multiplication mod 10.

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PARI

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A169963 Number of (2n+1)-digit squares in carryless arithmetic mod 10.

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Maple

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A361351 Carryless n-th powers of n base 10.

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