A006015 -id:A006015 - cp's OEIS frontend

A004468 a(n) = Nim product 3 * n.

0, 3, 1, 2, 12, 15, 13, 14, 4, 7, 5, 6, 8, 11, 9, 10, 48, 51, 49, 50, 60, 63, 61, 62, 52, 55, 53, 54, 56, 59, 57, 58, 16, 19, 17, 18, 28, 31, 29, 30, 20, 23, 21, 22, 24, 27, 25, 26, 32, 35, 33, 34, 44, 47, 45, 46, 36, 39, 37, 38, 40, 43, 41, 42, 192, 195, 193, 194, 204, 207, 205

Offset: 0

N. J. A. Sloane

From Jianing Song, Aug 10 2022: (Start)
Write n in quaternary (base 4), then replace each 1,2,3 by 3,1,2.
This is a permutation of the natural numbers; A006015 is the inverse permutation (since the nim product of 2 and 3 is 1). (End)

J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 51-53.

Alois P. Heinz, Table of n, a(n) for n = 0..16383 (first 1001 terms from R. J. Mathar)
Index entries for sequences related to Nim-multiplication
Index entries for sequences that are permutations of the natural numbers

Row 3 of array in A051775.

Cf. A006015, A004469-A004480, A000695, A062880, A001196.

read("transforms") ;
# insert Maple procedures nimprodP2() and A051775() of the b-file in A051775 here.
A004468 := proc(n)
        A051775(3,n) ;
end proc:
L := [seq(A004468(n),n=0..1000)] ; # R. J. Mathar, May 28 2011
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 0,
      a(iquo(n, 4, 'r'))*4+[0, 3, 1, 2][r+1])
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Jan 25 2022

a[n_] := a[n] = If[n == 0, 0, {q, r} = QuotientRemainder[n, 4]; a[q]*4 + {0, 3, 1, 2}[[r + 1]]];
Table[a[n], {n, 0, 70}] (* Jean-François Alcover, May 20 2022, after Alois P. Heinz *)

a(n) = my(v=digits(n, 4), w=[0,3,1,2]); for(i=1, #v, v[i] = w[v[i]+1]); fromdigits(v, 4) \\ Jianing Song, Aug 10 2022

def a(n, D=[0, 3, 1, 2]):
    r, k = 0, 0
    while n>0: r+=D[n%4]*4**k; n//=4; k+=1
    return r
# Onur Ozkan, Mar 07 2023

a(n) = A051775(3,n).
From Jianing Song, Aug 10 2022: (Start)
a(n) = 3*n if n has only digits 0 or 1 in quaternary (n is in A000695). Otherwise, a(n) < 3*n.
a(n) = n/2 if n has only digits 0 or 2 in quaternary (n is in A062880). Otherwise, a(n) > n/2.
a(n) = 2*n/3 if and only if n has only digits 0 or 3 in quaternary (n is in A001196). Proof: let n = Sum_i d_i*4^i, d(i) = 0,1,2,3. Write A = Sum_{d_i=1} 4^i, B = Sum_{d_i=2} 4^i, then a(n) = 2*n/3 if and only if 3*A + B = 2/3*(A + 2*B), or B = 7*A. If A != 0, then A is of the form (4*s+1)*4^t, but 7*A is not of this form. So the only possible case is A = B = 0, namely n has only digits 0 or 3. (End)

More terms from Erich Friedman

A334290 Array read by upward antidiagonals: T(n,k) (n > 0, k > 0) = nim-division of n by k.

Original entry on oeis.org

1, 2, 3, 3, 1, 2, 4, 2, 3, 15, 5, 12, 1, 5, 12, 6, 15, 8, 10, 4, 9, 7, 13, 10, 1, 8, 14, 11, 8, 14, 11, 14, 13, 7, 13, 10, 9, 4, 9, 4, 1, 15, 6, 15, 6, 10, 7, 12, 11, 9, 6, 7, 5, 11, 8, 11, 5, 14, 2, 5, 1, 12, 3, 13, 12, 7, 12, 6, 15, 13, 6, 8, 10, 9, 14, 4, 9, 5

Offset: 1

Rémy Sigrist, Jun 13 2020

Each row and each column is a permutation of the natural numbers.

			The array begins:
  n\k|   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
  ---+------------------------------------------------------------
    1|   1   3   2  15  12   9  11  10   6   8   7   5  14  13   4 --> A051917(n)
    2|   2   1   3   5   4  14  13  15  11  12   9  10   7   6   8
    3|   3   2   1  10   8   7   6   5  13   4  14  15   9  11  12
    4|   4  12   8   1  13  15   7   3  14  11  10   2   5   9   6
    5|   5  15  10  14   1   6  12   9   8   3  13   7  11   4   2
    6|   6  13  11   4   9   1  10  12   5   7   3   8   2  15  14
    7|   7  14   9  11   5   8   1   6   3  15   4  13  12   2  10
    8|   8   4  12   2   6   5   9   1   7  13  15   3  10  14  11
    9|   9   7  14  13  10  12   2  11   1   5   8   6   4   3  15
   10|  10   5  15   7   2  11   4  14  12   1   6   9  13   8   3
   11|  11   6  13   8  14   2  15   4  10   9   1  12   3   5   7
   12|  12   8   4   3  11  10  14   2   9   6   5   1  15   7  13
   13|  13  11   6  12   7   3   5   8  15  14   2   4   1  10   9
   14|  14   9   7   6  15   4   3  13   2  10  12  11   8   1   5
   15|  15  10   5   9   3  13   8   7   4   2  11  14   6  12   1
             |   |   |   |   |
             |   |   |   |   A004474(n)
             |   |   |   A004477(n)
             |   |   A004480(n)
             |   A006015(n)
             A004468(n)

Rémy Sigrist, Table of n, a(n) for n = 1..5050
Rémy Sigrist, PARI program for A334290
Index entries for sequences related to Nim-multiplication

Cf. A004468, A004474, A004477, A004480, A006015, A051775, A051776, A051917, A334291.

PARI
```
See Links section.
```

T(n, k) = A051776(n, A051917(k)).
T(n, 1) = n.
T(1, n) = A051917(k).
T(n, n) = 1.

A334291 Array read by upward antidiagonals: T(n,k) (n >= 0, k > 0) = nim-division of n by k.

Original entry on oeis.org

0, 1, 0, 2, 3, 0, 3, 1, 2, 0, 4, 2, 3, 15, 0, 5, 12, 1, 5, 12, 0, 6, 15, 8, 10, 4, 9, 0, 7, 13, 10, 1, 8, 14, 11, 0, 8, 14, 11, 14, 13, 7, 13, 10, 0, 9, 4, 9, 4, 1, 15, 6, 15, 6, 0, 10, 7, 12, 11, 9, 6, 7, 5, 11, 8, 0, 11, 5, 14, 2, 5, 1, 12, 3, 13, 12, 7

Offset: 0

Rémy Sigrist, Jun 13 2020

This is the array A334290 with a leading row of 0's.

			The array begins:
  n\k|   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
  ---+------------------------------------------------------------
    0|   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0
    1|   1   3   2  15  12   9  11  10   6   8   7   5  14  13   4 --> A051917(n)
    2|   2   1   3   5   4  14  13  15  11  12   9  10   7   6   8
    3|   3   2   1  10   8   7   6   5  13   4  14  15   9  11  12
    4|   4  12   8   1  13  15   7   3  14  11  10   2   5   9   6
    5|   5  15  10  14   1   6  12   9   8   3  13   7  11   4   2
    6|   6  13  11   4   9   1  10  12   5   7   3   8   2  15  14
    7|   7  14   9  11   5   8   1   6   3  15   4  13  12   2  10
    8|   8   4  12   2   6   5   9   1   7  13  15   3  10  14  11
    9|   9   7  14  13  10  12   2  11   1   5   8   6   4   3  15
   10|  10   5  15   7   2  11   4  14  12   1   6   9  13   8   3
   11|  11   6  13   8  14   2  15   4  10   9   1  12   3   5   7
   12|  12   8   4   3  11  10  14   2   9   6   5   1  15   7  13
   13|  13  11   6  12   7   3   5   8  15  14   2   4   1  10   9
   14|  14   9   7   6  15   4   3  13   2  10  12  11   8   1   5
   15|  15  10   5   9   3  13   8   7   4   2  11  14   6  12   1
             |   |   |   |   |
             |   |   |   |   A004474(n)
             |   |   |   A004477(n)
             |   |   A004480(n)
             |   A006015(n)
             A004468(n)

Index entries for sequences related to Nim-multiplication

Cf. A004468, A004474, A004477, A004480, A006015, A051775, A051776, A051917, A334290.

T(n, k) = A051775(n, A051917(k)).
T(n, 1) = n.
T(1, n) = A051917(k).
T(n, n) = 1.

cp's OEIS Frontend

A004468 a(n) = Nim product 3 * n.

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A334290 Array read by upward antidiagonals: T(n,k) (n > 0, k > 0) = nim-division of n by k.

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A334291 Array read by upward antidiagonals: T(n,k) (n >= 0, k > 0) = nim-division of n by k.

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Author

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Comments

Examples

Links

Crossrefs

Formula