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 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS From Jianing Song, Aug 10 2022: (Start) 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. 51-53. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS 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 FORMULA From Jianing Song, Aug 10 2022: (Start) a(n) = A051775(2,n). 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: seq(a(n), n=0..70); # Alois P. Heinz, Jan 25 2022 MATHEMATICA a[n_] := a[n] = If[n == 0, 0, {q, r} = QuotientRemainder[n, 4]; a[q]*4 + {0, 2, 3, 1}[[r + 1]]]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, May 20 2022, after Alois P. Heinz *) 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 # Onur Ozkan, Mar 07 2023 CROSSREFS Row 2 of array in A051775. Cf. A004468-A004480, A000695, A062880, A001196. Sequence in context: A101281 A106033 A121634 * A301332 A110292 A138672 Adjacent sequences: A006012 A006013 A006014 * A006016 A006017 A006018 KEYWORD nonn,easy,look AUTHOR N. J. A. Sloane. EXTENSIONS More terms from Erich Friedman. STATUS approved

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Last modified December 5 14:39 EST 2023. Contains 367592 sequences. (Running on oeis4.)