A055115 -id:A055115 - cp's OEIS frontend

A004488 Tersum n + n.

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

A055120 Digital complement of n (replace each nonzero digit d with 10-d).

Original entry on oeis.org

0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 90, 99, 98, 97, 96, 95, 94, 93, 92, 91, 80, 89, 88, 87, 86, 85, 84, 83, 82, 81, 70, 79, 78, 77, 76, 75, 74, 73, 72, 71, 60, 69, 68, 67, 66, 65, 64, 63, 62, 61, 50, 59, 58, 57, 56, 55, 54, 53, 52, 51, 40, 49, 48, 47, 46, 45, 44, 43, 42, 41, 30, 39

Offset: 0

Henry Bottomley, Apr 19 2000

a(n) = -n in carryless arithmetic mod 10 - that is, n + a(n) = 0 (cf. A169894). - N. J. A. Sloane, Aug 03 2010

			a(11) = 99 because 1 + 9 = 0 mod 10 for each digit.
a(20) = 80 because 2 + 8 = 0 mod 10 and 0 + 0 = 0 mod 10.

N. J. A. Sloane, 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.
Index entries for sequences related to carryless arithmetic
Index entries for sequences that are permutations of the natural numbers

Cf. A004488, A048647, A055115-A055126, A061601.

Column k=10 of A248813.

a055120 = foldl f 0 . reverse . unfoldr g where
   f v d = if d == 0 then 10 * v else 10 * v + 10 - d
   g x = if x == 0 then Nothing else Just $ swap $ divMod x 10
-- Reinhard Zumkeller, Oct 04 2011

f:=proc(n) local t0,t1,i;
t0:=0; t1:=convert(n,base,10);
for i from 1 to nops(t1) do
if t1[i]>0 then t0:=t0+(10-t1[i])*10^(i-1); fi;
od:
RETURN(t0);
end;
# N. J. A. Sloane, Jan 21 2011

a[n_] := FromDigits[ IntegerDigits[n] /. d_?Positive -> 10-d]; Table[a[n], {n, 0, 100}](* Jean-François Alcover, Nov 28 2011 *)

a(n)=fromdigits(apply(d->if(d,10-d,0),digits(n))) \\ Charles R Greathouse IV, Feb 08 2017

def A055120(n): return int(''.join(str(10-int(d)) if d != '0' else d for d in str(n))) # Chai Wah Wu, Apr 03 2021

From Robert Israel, Sep 04 2017: (Start)
a(10*n) = 10*a(n).
a(10*n+j) = 10*a(n) + 10 - j for 1 <= j <= 9.
G.f. g(x) satisfies g(x) = 10*(1+x+x^2+...+x^9)*g(x^10) + (9*x+8*x^2+7*x^3+6*x^4+5*x^5+4*x^6+3*x^7+2*x^8+x^9)/(1-x^10).
(End)

A248813 A(n,k) is the base-k complement of n; square array A(n,k), n>=0, k>=2, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 0, 3, 1, 3, 0, 4, 2, 6, 4, 0, 5, 3, 1, 8, 5, 0, 6, 4, 2, 12, 7, 6, 0, 7, 5, 3, 1, 15, 3, 7, 0, 8, 6, 4, 2, 20, 14, 5, 8, 0, 9, 7, 5, 3, 1, 24, 13, 4, 9, 0, 10, 8, 6, 4, 2, 30, 23, 8, 18, 10, 0, 11, 9, 7, 5, 3, 1, 35, 22, 11, 20, 11, 0, 12, 10, 8, 6, 4, 2, 42, 34, 21, 10, 19, 12

Offset: 0

Alois P. Heinz, Mar 03 2015

Every column is a permutation of the nonnegative integers.

			Square array A(n,k) begins:
   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, ...
   1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, ...
   2,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, ...
   3,  6,  1,  2,  3,  4,  5,  6,  7,  8,  9, ...
   4,  8, 12,  1,  2,  3,  4,  5,  6,  7,  8, ...
   5,  7, 15, 20,  1,  2,  3,  4,  5,  6,  7, ...
   6,  3, 14, 24, 30,  1,  2,  3,  4,  5,  6, ...
   7,  5, 13, 23, 35, 42,  1,  2,  3,  4,  5, ...
   8,  4,  8, 22, 34, 48, 56,  1,  2,  3,  4, ...
   9, 18, 11, 21, 33, 47, 63, 72,  1,  2,  3, ...
  10, 20, 10, 15, 32, 46, 62, 80, 90,  1,  2, ...

Columns k=2-16 give: A001477, A004488, A048647, A055115, A055116, A055117, A055118, A055119, A055120, A055121, A055122, A055123, A055124, A055125, A055126.

A:= proc(n, k) local t, r, i; t, r:= n, 0;
      for i from 0 while t>0 do
        r:= r+k^i *irem(k-irem(t, k, 't'), k)
      od; r
    end:
seq(seq(A(n, 2+d-n), n=0..d), d=0..14);

A(n,k)=fromdigits(apply(d->(k-d)%k, digits(n, k)), k); \\ Gheorghe Coserea, Apr 23 2018

A055126 Base-16 complement of n (write n in base 16, then replace each digit with its base-16 negative).

Original entry on oeis.org

0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 240, 255, 254, 253, 252, 251, 250, 249, 248, 247, 246, 245, 244, 243, 242, 241, 224, 239, 238, 237, 236, 235, 234, 233, 232, 231, 230, 229, 228, 227, 226, 225, 208, 223, 222, 221, 220, 219, 218, 217, 216, 215

Offset: 0

Henry Bottomley, Apr 19 2000

Reinhard Zumkeller, Table of n, a(n) for n = 0..4095 (4095 = 16^3 - 1)
Index entries for sequences that are permutations of the natural numbers

Cf. A004488, A048647, A055115-A055126.

Column k=16 of A248813.

a055126 0 = 0
a055126 n = if d == 0 then 16 * a055126 n' else 16 * a055126 n' + 16 - d
            where (n', d) = divMod n 16
-- Reinhard Zumkeller, Mar 12 2014

A055116 Base-6 complement of n (write n in base 6, then replace each digit with its base-6 negative).

Original entry on oeis.org

0, 5, 4, 3, 2, 1, 30, 35, 34, 33, 32, 31, 24, 29, 28, 27, 26, 25, 18, 23, 22, 21, 20, 19, 12, 17, 16, 15, 14, 13, 6, 11, 10, 9, 8, 7, 180, 185, 184, 183, 182, 181, 210, 215, 214, 213, 212, 211, 204, 209, 208, 207, 206, 205, 198, 203, 202, 201, 200, 199, 192, 197, 196

Offset: 0

Henry Bottomley, Apr 19 2000

Cf. A004488, A048647, A055115-A055126.

Column k=6 of A248813.

A055117 Base-7 complement of n (write n in base 7, then replace each digit with its base-7 negative).

Original entry on oeis.org

0, 6, 5, 4, 3, 2, 1, 42, 48, 47, 46, 45, 44, 43, 35, 41, 40, 39, 38, 37, 36, 28, 34, 33, 32, 31, 30, 29, 21, 27, 26, 25, 24, 23, 22, 14, 20, 19, 18, 17, 16, 15, 7, 13, 12, 11, 10, 9, 8, 294, 300, 299, 298, 297, 296, 295, 336, 342, 341, 340, 339, 338, 337, 329, 335, 334

Offset: 0

Henry Bottomley, Apr 19 2000

Cf. A004488, A048647, A055115-A055126.

Column k=7 of A248813.

A055118 Base-8 complement of n (write n in base 8, then replace each digit with its base-8 negative).

Original entry on oeis.org

0, 7, 6, 5, 4, 3, 2, 1, 56, 63, 62, 61, 60, 59, 58, 57, 48, 55, 54, 53, 52, 51, 50, 49, 40, 47, 46, 45, 44, 43, 42, 41, 32, 39, 38, 37, 36, 35, 34, 33, 24, 31, 30, 29, 28, 27, 26, 25, 16, 23, 22, 21, 20, 19, 18, 17, 8, 15, 14, 13, 12, 11, 10, 9, 448, 455, 454, 453, 452, 451

Offset: 0

Henry Bottomley, Apr 19 2000

Reinhard Zumkeller, Table of n, a(n) for n = 0..4095 (4095 = 8^4 - 1)
Index entries for sequences that are permutations of the natural numbers

Cf. A004488, A048647, A055115-A055126.

Column k=8 of A248813.

a055118 0 = 0
a055118 n = if d == 0 then 8 * a055118 n' else 8 * a055118 n' + 8 - d
            where (n', d) = divMod n 8
-- Reinhard Zumkeller, Mar 12 2014

A055119 Base-9 complement of n (write n in base 9, then replace each digit with its base-9 negative).

Original entry on oeis.org

0, 8, 7, 6, 5, 4, 3, 2, 1, 72, 80, 79, 78, 77, 76, 75, 74, 73, 63, 71, 70, 69, 68, 67, 66, 65, 64, 54, 62, 61, 60, 59, 58, 57, 56, 55, 45, 53, 52, 51, 50, 49, 48, 47, 46, 36, 44, 43, 42, 41, 40, 39, 38, 37, 27, 35, 34, 33, 32, 31, 30, 29, 28, 18, 26, 25, 24, 23, 22, 21, 20, 19

Offset: 0

Henry Bottomley, Apr 19 2000

Cf. A004488, A048647, A055115-A055126.

Column k=9 of A248813.

A055121 Base-11 complement of n (write n in base 11, then replace each digit with its base-11 negative).

Original entry on oeis.org

0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 110, 120, 119, 118, 117, 116, 115, 114, 113, 112, 111, 99, 109, 108, 107, 106, 105, 104, 103, 102, 101, 100, 88, 98, 97, 96, 95, 94, 93, 92, 91, 90, 89, 77, 87, 86, 85, 84, 83, 82, 81, 80, 79, 78, 66, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67

Offset: 0

Henry Bottomley, Apr 19 2000

Cf. A004488, A048647, A055115-A055126.

Column k=11 of A248813.

A055122 Base-12 complement of n (write n in base 12, then replace each digit with its base-12 negative).

Original entry on oeis.org

0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 132, 143, 142, 141, 140, 139, 138, 137, 136, 135, 134, 133, 120, 131, 130, 129, 128, 127, 126, 125, 124, 123, 122, 121, 108, 119, 118, 117, 116, 115, 114, 113, 112, 111, 110, 109, 96, 107, 106, 105, 104, 103, 102, 101, 100, 99

Offset: 0

Henry Bottomley, Apr 19 2000

Reinhard Zumkeller, Table of n, a(n) for n = 0..20735 (20735 = 12^4 - 1)
Index entries for sequences that are permutations of the natural numbers

Cf. A004488, A048647, A055115-A055126.

Column k=12 of A248813.

a055122 0 = 0
a055122 n = if d == 0 then 12 * a055122 n' else 12 * a055122 n' + 12 - d
            where (n', d) = divMod n 12
-- Reinhard Zumkeller, Mar 12 2014

cp's OEIS Frontend

A004488 Tersum n + n.

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A055120 Digital complement of n (replace each nonzero digit d with 10-d).

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A248813 A(n,k) is the base-k complement of n; square array A(n,k), n>=0, k>=2, read by antidiagonals.

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A055126 Base-16 complement of n (write n in base 16, then replace each digit with its base-16 negative).

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A055116 Base-6 complement of n (write n in base 6, then replace each digit with its base-6 negative).

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A055117 Base-7 complement of n (write n in base 7, then replace each digit with its base-7 negative).

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A055118 Base-8 complement of n (write n in base 8, then replace each digit with its base-8 negative).

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A055119 Base-9 complement of n (write n in base 9, then replace each digit with its base-9 negative).

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A055121 Base-11 complement of n (write n in base 11, then replace each digit with its base-11 negative).

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