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A036668 Hati numbers: of form 2^i*3^j*k, i+j even, (k,6)=1.

%I A036668 #52 Feb 12 2025 11:55:30
%S A036668 1,4,5,6,7,9,11,13,16,17,19,20,23,24,25,28,29,30,31,35,36,37,41,42,43,
%T A036668 44,45,47,49,52,53,54,55,59,61,63,64,65,66,67,68,71,73,76,77,78,79,80,
%U A036668 81,83,85,89,91,92,95,96,97,99,100,101,102,103,107
%N A036668 Hati numbers: of form 2^i*3^j*k, i+j even, (k,6)=1.
%C A036668 If n appears then 2n and 3n do not. - _Benoit Cloitre_, Jun 13 2002
%C A036668 Closed under multiplication. Each term is a product of a unique subset of {6} U A050376 \ {2,3}. - _Peter Munn_, Sep 14 2019
%H A036668 Robert Israel, <a href="/A036668/b036668.txt">Table of n, a(n) for n = 0..10000</a>
%H A036668 Don McDonald, <a href="https://groups.google.com/g/sci.math/c/o8pXfju6UQ4/m/sXwtqHfqTncJ">Obituary of Alan Robert Boyd</a>, posted to sci.math Jan 02 1999; <a href="http://nzsm.webcentre.co.nz/article1926.htm">alternative link</a>.
%F A036668 a(n) = 12/7 * n + O(log^2 n). - _Charles R Greathouse IV_, Sep 10 2015
%F A036668 {a(n)} = A052330({A014601(n)}), where {a(n)} denotes the set of integers in the sequence. - _Peter Munn_, Sep 14 2019
%p A036668 N:= 1000: # to get all terms up to N
%p A036668 A:= {seq(2^i,i=0..ilog2(N))}:
%p A036668 Ae,Ao:= selectremove(issqr,A):
%p A036668 Be:= map(t -> seq(t*9^j, j=0 .. floor(log[9](N/t))),Ae):
%p A036668 Bo:= map(t -> seq(t*3*9^j,j=0..floor(log[9](N/(3*t)))),Ao):
%p A036668 B:= Be union Bo:
%p A036668 C1:= map(t -> seq(t*(6*i+1),i=0..floor((N/t -1)/6)),B):
%p A036668 C2:= map(t -> seq(t*(6*i+5),i=0..floor((N/t - 5)/6)),B):
%p A036668 A036668:= C1 union C2; # _Robert Israel_, May 09 2014
%t A036668 a = {1}; Do[AppendTo[a, NestWhile[# + 1 &, Last[a] + 1,
%t A036668 Apply[Or, Map[MemberQ[a, #] &, Select[Flatten[{#/3, #/2}],
%t A036668 IntegerQ]]] &]], {150}]; a  (* A036668 *)
%t A036668 (* _Peter J. C. Moses_, Apr 23 2019 *)
%o A036668 (PARI) twos(n) = {local(r,m);r=0;m=n;while(m%2==0,m=m/2;r++);r}
%o A036668 threes(n) = {local(r,m);r=0;m=n;while(m%3==0,m=m/3;r++);r}
%o A036668 isA036668(n) = (twos(n)+threes(n))%2==0 \\ _Michael B. Porter_, Mar 16 2010
%o A036668 (PARI) is(n)=(valuation(n,2)+valuation(n,3))%2==0 \\ _Charles R Greathouse IV_, Sep 10 2015
%o A036668 (PARI) list(lim)=my(v=List(),N);for(n=0,logint(lim\=1,3),N=if(n%2,2*3^n,3^n); while(N<=lim, forstep(k=N,lim,[4*N,2*N], listput(v,k)); N<<=2)); Set(v) \\ _Charles R Greathouse IV_, Sep 10 2015
%o A036668 (Python)
%o A036668 from itertools import count
%o A036668 def A036668(n):
%o A036668     def f(x):
%o A036668         c = n+x
%o A036668         for i in range(x.bit_length()+1):
%o A036668             i2 = 1<<i
%o A036668             for j in count(i&1,2):
%o A036668                 k = i2*3**j
%o A036668                 if k>x:
%o A036668                     break
%o A036668                 m = x//k
%o A036668                 c -= (m-1)//6+(m-5)//6+2
%o A036668         return c
%o A036668     m, k = n, f(n)
%o A036668     while m != k: m, k = k, f(k)
%o A036668     return m # _Chai Wah Wu_, Jan 28 2025
%Y A036668 Cf. A003159, A007310, A014601, A036667, A050376, A052330, A325424 (complement), A325498 (first differences), A373136 (characteristic function).
%Y A036668 Positions of 0's in A182582.
%Y A036668 Subsequences: A084087, A339690, A352272, A352273.
%Y A036668 Cf. also A121537, A145204, A225837, A307150, A329575, A334747, A334748, A339746.
%K A036668 nonn,easy
%O A036668 1,2
%A A036668 _N. J. A. Sloane_, Antreas P. Hatzipolakis (xpolakis(AT)hol.gr)
%E A036668 Offset changed by _Chai Wah Wu_, Jan 28 2025