A242399 -id:A242399 - cp's OEIS frontend

A242400 Differences between A008586 (multiples of 4) and A242399.

0, 0, 0, 0, 0, 9, 0, 9, 9, 0, 0, 0, 0, 0, 9, 27, 36, 36, 0, 0, 0, 27, 27, 36, 27, 36, 36, 0, 0, 0, 0, 0, 9, 0, 9, 9, 0, 0, 0, 0, 0, 9, 27, 36, 36, 81, 81, 81, 108, 108, 117, 108, 117, 117, 0, 0, 0, 0, 0, 9, 0, 9, 9, 81, 81, 81, 81, 81, 90, 108, 117, 117, 81

Offset: 0

Reinhard Zumkeller, May 13 2014

nonn

a(n) = A008586(n) - A242399(n);
a(m) = 0 iff m is a term of A242407;
a(A242407(n)) = 0; a(A242408(n)) > 0.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A048728.

Haskell
```
a242400 n = a008586 n - a242399 n
```

A048724 Write n and 2n in binary and add them mod 2.

Original entry on oeis.org

0, 3, 6, 5, 12, 15, 10, 9, 24, 27, 30, 29, 20, 23, 18, 17, 48, 51, 54, 53, 60, 63, 58, 57, 40, 43, 46, 45, 36, 39, 34, 33, 96, 99, 102, 101, 108, 111, 106, 105, 120, 123, 126, 125, 116, 119, 114, 113, 80, 83, 86, 85, 92, 95, 90, 89, 72, 75, 78, 77, 68, 71, 66, 65, 192

Offset: 0

Antti Karttunen, Apr 26 1999

Reversing binary representation of -n. Converting sum of powers of 2 in binary representation of a(n) to alternating sum gives -n. Note that the alternation is applied only to the nonzero bits and does not depend on the exponent of two. All integers have a unique reversing binary representation (see cited exercise for proof). Complement of A065621. - Marc LeBrun, Nov 07 2001
A permutation of the "evil" numbers A001969. - Marc LeBrun, Nov 07 2001
A048725(n) = a(a(n)). - Reinhard Zumkeller, Nov 12 2004

			12 = 1100 in binary, 24=11000 and their sum is 10100=20, so a(12)=20.
a(4) = 12 = + 8 + 4 -> - 8 + 4 = -4.

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1969, Vol. 2, p. 178, (exercise 4.1. Nr. 27)

T. D. Noe, Table of n, a(n) for n = 0..1023
P. Mathonet, M. Rigo, M. Stipulanti and N. Zénaïdi, On digital sequences associated with Pascal's triangle, arXiv:2201.06636 [math.NT], 2022.
H. D. Nguyen, A mixing of Prouhet-Thue-Morse sequences and Rademacher functions, 2014. See Example 20. - _N. J. A. Sloane_, May 24 2014
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions

Cf. A048720, A048725, A048726, A048728.
Bisection of A003188 (even part).
See also A065620, A065621.
Cf. A242399.

import Data.Bits (xor, shiftL)
a048724 n = n `xor` shiftL n 1 :: Integer
-- Reinhard Zumkeller, Mar 06 2013

a:= n-> Bits[Xor](n, n+n):
seq(a(n), n=0..100);  # Alois P. Heinz, Apr 06 2016

Table[ BitXor[2n, n], {n, 0, 65}] (* Robert G. Wilson v, Jul 06 2006 *)

a(n)=bitxor(n,2*n) \\ Charles R Greathouse IV, Jan 04 2013

def A048724(n): return n^(n<<1) # Chai Wah Wu, Apr 05 2021

a(n) = Xmult(n, 3) (or n XOR (n<<1)).
a(n) = A065621(-n).
a(2n) = 2a(n), a(2n+1) = 2a(n) + 2(-1)^n + 1.
G.f. 1/(1-x) * sum(k>=0, 2^k*(3t-t^3)/(1+t)/(1+t^2), t=x^2^k). - Ralf Stephan, Sep 08 2003
a(n) = sum(k=0, n, (1-(-1)^round(+n/2^k))/2*2^k). - Benoit Cloitre, Apr 27 2005
a(n) = A001969(A003188(n)). - Philippe Deléham, Apr 29 2005
a(n) = A106409(2*n) for n>0. - Reinhard Zumkeller, May 02 2005
a(n) = A142149(2*n). - Reinhard Zumkeller, Jul 15 2008

A004488 Tersum n + n.

Original entry on oeis.org

0, 2, 1, 6, 8, 7, 3, 5, 4, 18, 20, 19, 24, 26, 25, 21, 23, 22, 9, 11, 10, 15, 17, 16, 12, 14, 13, 54, 56, 55, 60, 62, 61, 57, 59, 58, 72, 74, 73, 78, 80, 79, 75, 77, 76, 63, 65, 64, 69, 71, 70, 66, 68, 67, 27, 29, 28, 33, 35, 34, 30, 32, 31, 45, 47, 46, 51

Offset: 0

N. J. A. Sloane

Could also be described as "Write n in base 3, then replace each digit with its base-3 negative" as with A048647 for base 4. - Henry Bottomley, Apr 19 2000
a(a(n)) = n, a self-inverse permutation of the nonnegative integers. - Reinhard Zumkeller, Dec 19 2003
First 3^n terms of the sequence form a permutation s(n) of 0..3^n-1, n>=1; the number of inversions of s(n) is A016142(n-1). - Gheorghe Coserea, Apr 23 2018

Gheorghe Coserea, Table of n, a(n) for n = 0..59048 (first 6561 terms from Alois P. Heinz)
Index entries for sequences related to carryless arithmetic
Index entries for sequences that are permutations of the natural numbers

Column k=0 of A253586, A253587.
Column k=3 of A248813.
Row / column 2 of A325820.
Main diagonal of A004489.
Cf. A048647, A055115, A055116, A055120, A059249, A117966, A117967, A117968, A225901, A242399, A244042, A263273, A289813, A289814, A289815, A289816, A289831, A289838, A300222, A321464.

a004488 0 = 0
a004488 n = if d == 0 then 3 * a004488 n' else 3 * a004488 n' + 3 - d
            where (n', d) = divMod n 3
-- Reinhard Zumkeller, Mar 12 2014

a:= proc(n) local t, r, i;
      t, r:= n, 0;
      for i from 0 while t>0 do
        r:= r+3^i *irem(2*irem(t, 3, 't'), 3)
      od; r
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Sep 07 2011

a[n_] := FromDigits[Mod[3-IntegerDigits[n, 3], 3], 3]; Table[a[n], {n, 0, 66}] (* Jean-François Alcover, Mar 03 2014 *)

a(n) = my(b=3); fromdigits(apply(d->(b-d)%b, digits(n, b)), b);
vector(67, i, a(i-1))  \\ Gheorghe Coserea, Apr 23 2018

from sympy.ntheory.factor_ import digits
def a(n): return int("".join([str((3 - i)%3) for i in digits(n, 3)[1:]]), 3) # Indranil Ghosh, Jun 06 2017

Tersum m + n: write m and n in base 3 and add mod 3 with no carries, e.g., 5 + 8 = "21" + "22" = "10" = 1.
a(n) = Sum(3-d(i)-3*0^d(i): n=Sum(d(i)*3^d(i): 0<=d(i)<3)). - Reinhard Zumkeller, Dec 19 2003
a(3*n) = 3*a(n), a(3*n+1) = 3*a(n)+2, a(3*n+2) = 3*a(n)+1. - Robert Israel, May 09 2014

A325820 Multiplication table for carryless product i X j in base 3 for i >= 0 and j >= 0, read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 1, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 8, 9, 8, 5, 0, 0, 6, 7, 12, 12, 7, 6, 0, 0, 7, 3, 15, 16, 15, 3, 7, 0, 0, 8, 5, 18, 11, 11, 18, 5, 8, 0, 0, 9, 4, 21, 24, 13, 24, 21, 4, 9, 0, 0, 10, 18, 24, 19, 21, 21, 19, 24, 18, 10, 0, 0, 11, 20, 27, 23, 26, 9, 26, 23, 27, 20, 11, 0, 0, 12, 19, 30, 36, 19, 15, 15, 19, 36, 30, 19, 12, 0

Offset: 0

Antti Karttunen, May 22 2019

			The array begins as:
  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, ...
  0,  2,  1,  6,  8,  7,  3,  5,  4,  18,  20,  19,  24, ...
  0,  3,  6,  9, 12, 15, 18, 21, 24,  27,  30,  33,  36, ...
  0,  4,  8, 12, 16, 11, 24, 19, 23,  36,  40,  44,  48, ...
  0,  5,  7, 15, 11, 13, 21, 26, 19,  45,  50,  52,  33, ...
  0,  6,  3, 18, 24, 21,  9, 15, 12,  54,  60,  57,  72, ...
  0,  7,  5, 21, 19, 26, 15, 13, 11,  63,  70,  68,  57, ...
  0,  8,  4, 24, 23, 19, 12, 11, 16,  72,  80,  76,  69, ...
  0,  9, 18, 27, 36, 45, 54, 63, 72,  81,  90,  99, 108, ...
  0, 10, 20, 30, 40, 50, 60, 70, 80,  90, 100,  83, 120, ...
  0, 11, 19, 33, 44, 52, 57, 68, 76,  99,  83,  91, 132, ...
  0, 12, 24, 36, 48, 33, 72, 57, 69, 108, 120, 132, 144, ...
  etc.
A(2,2) = 2*2 mod 3 = 1.

Cf. A169999 (the main diagonal).
Row/Column 0: A000004, Row/Column 1: A001477, Row/Column 2: A004488, Row/Column 3: A008585, Row/Column 4: A242399, Row/Column 9: A008591.
Cf. A007089, A207669, A207670, A263273, A325825.
Cf. A325821 (same table without the zero row and column).
Cf. A048720 (binary), A059692 (decimal), A004247 (full multiply).

up_to = 105;
A325820sq(b, c) = fromdigits(Vec(Pol(digits(b,3))*Pol(digits(c,3)))%3, 3);
A325820list(up_to) = { my(v = vector(up_to), i=0); for(a=0,oo, for(col=0,a, if(i++ > up_to, return(v)); v[i] = A325820sq(a-col,col))); (v); };
v325820 = A325820list(up_to);
A325820(n) = v325820[1+n];

A242407 Numbers such that in ternary representation all pairs of adjacent digits have sums not greater than 2.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 9, 10, 11, 12, 13, 18, 19, 20, 27, 28, 29, 30, 31, 33, 36, 37, 38, 39, 40, 54, 55, 56, 57, 58, 60, 81, 82, 83, 84, 85, 87, 90, 91, 92, 93, 94, 99, 100, 101, 108, 109, 110, 111, 112, 114, 117, 118, 119, 120, 121, 162, 163, 164, 165, 166, 168

Offset: 1

Reinhard Zumkeller, May 13 2014

A242400(a(n)) = 0;
A242399(a(n)) = 4*a(n);
numbers m, such that in ternary arithmetic no carry occurs, when 3*m is added to m.

			Initial terms and their ternary representations, cf. A007089:
.  0 1 2  3  4  6   9  10  11  12  13  18  19  20   27   28   29   30 ..
.  0 1 2 10 11 20 100 101 102 110 111 200 201 202 1000 1001 1002 1010 ..

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Ternary.
Wikipedia, Ternary numeral system

Cf. A242408 (complement), A003714, A039691, A007089.

a242407 n = a242407_list !! (n-1)
a242407_list = filter ((== 0) . a242400) [0..]

Select[Range[0,200],Max[Total/@Partition[IntegerDigits[#, 3],2,1]]<3&] (* Harvey P. Dale, Jan 08 2023 *)

A242408 Numbers such that in ternary representation at least one pair of adjacent digits has a sum greater than 2.

Original entry on oeis.org

5, 7, 8, 14, 15, 16, 17, 21, 22, 23, 24, 25, 26, 32, 34, 35, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 86, 88, 89, 95, 96, 97, 98, 102, 103, 104, 105, 106, 107, 113

Offset: 1

Reinhard Zumkeller, May 13 2014

A242400(a(n)) > 0;

A242399(a(n)) < 4*a(n).

			Initial terms and their ternary representations, cf. A007089:
.  5  7  8  14  15  16  17  21  22  23  24  25  26   32   34   35   41 ..
. 12 21 22 112 120 121 122 210 211 212 220 221 222 1012 1021 1022 1112 ..

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Ternary.
Wikipedia, Ternary numeral system

Cf. A242407 (complement), A004780.

a242408 n = a242408_list !! (n-1)
a242408_list = filter ((> 0) . a242400) [0..]

Select[Range[200],Max[Total/@Partition[IntegerDigits[#,3],2,1]]>2&] (* Harvey P. Dale, Feb 12 2016 *)

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

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A242400 Differences between A008586 (multiples of 4) and A242399.

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A048724 Write n and 2n in binary and add them mod 2.

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A004488 Tersum n + n.

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A325820 Multiplication table for carryless product i X j in base 3 for i >= 0 and j >= 0, read by antidiagonals.

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A242407 Numbers such that in ternary representation all pairs of adjacent digits have sums not greater than 2.

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Mathematica

A242408 Numbers such that in ternary representation at least one pair of adjacent digits has a sum greater than 2.

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Haskell

Mathematica

A059632 Carryless product 11 X n base 10.

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