

A006015


Nim product 2*n.
(Formerly M0412)


4



0, 2, 3, 1, 8, 10, 11, 9, 12, 14, 15, 13, 4, 6, 7, 5, 32, 34, 35, 33, 40, 42, 43, 41, 44, 46, 47, 45, 36, 38, 39, 37, 48, 50, 51, 49, 56, 58, 59, 57, 60, 62, 63, 61, 52, 54, 55, 53, 16, 18, 19, 17, 24, 26, 27, 25, 28, 30, 31, 29, 20, 22, 23, 21, 128, 130, 131, 129, 136, 138, 139
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OFFSET

0,2


COMMENTS

Write n in quaternary (base 4), then replace each 1,2,3 by 2,3,1.
This is a permutation of the natural numbers; A004468 is the inverse permutation (since the nim product of 2 and 3 is 1). (End)


REFERENCES

J. H. Conway, On Numbers and Games. Academic Press, NY, 1976, pp. 5153.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS



FORMULA

a(n) = 2*n if n has only digits 0 or 1 in quaternary (n is in A000695). Otherwise, a(n) < 2*n.
a(n) = n/3 if n has only digits 0 or 3 in quaternary (n is in A001196). Otherwise, a(n) > n/3.
a(n) = 3*n/2 if and only if n has only digits 0 or 2 in quaternary (n is in A062880). 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=3} 4^i, then a(n) = 3*n/2 if and only if 2*A + B = 3/2*(A + 3*B), or A = 7*B. If B != 0, then B is of the form (4*s+1)*4^t, but 7*B is not of this form. So the only possible case is A = B = 0, namely n has only digits 0 or 2. (End)


MAPLE

a:= proc(n) option remember; `if`(n=0, 0,
a(iquo(n, 4, 'r'))*4+[0, 2, 3, 1][r+1])
end:


MATHEMATICA

a[n_] := a[n] = If[n == 0, 0, {q, r} = QuotientRemainder[n, 4]; a[q]*4 + {0, 2, 3, 1}[[r + 1]]];


PROG

(PARI) a(n) = my(v=digits(n, 4), w=[0, 2, 3, 1]); for(i=1, #v, v[i] = w[v[i]+1]); fromdigits(v, 4) \\ Jianing Song, Aug 10 2022
(Python)
def a(n, D=[0, 2, 3, 1]):
r, k = 0, 0
while n>0: r+=D[n%4]*4**k; n//=4; k+=1
return r


CROSSREFS



KEYWORD



AUTHOR



EXTENSIONS



STATUS

approved



