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Minimal increasing sequence with a(1)=3 and the property that a(n) and n are both in or both not in A003159.
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%I #15 Dec 15 2018 08:34:08

%S 3,6,7,9,11,14,15,18,19,22,23,25,27,30,31,33,35,38,39,41,43,46,47,50,

%T 51,54,55,57,59,62,63,66,67,70,71,73,75,78,79,82,83,86,87,89,91,94,95,

%U 97,99,102,103,105,107,110,111,114,115,118,119,121,123,126,127,129,131,134

%N Minimal increasing sequence with a(1)=3 and the property that a(n) and n are both in or both not in A003159.

%C The primes in this sequence give A160216.

%C Conjecture: Let m>3 belong to A003159. Define the sequence b(n) to be the minimal increasing sequence with b(1)=m and the property that b(n) and n are both in or both not in A003159. Then a(n)=b(n) for all n larger than some m-dependent minimum index.

%H V. Shevelev, <a href="https://arxiv.org/abs/0904.2101">Several results on sequences which are similar to the positive integers</a>, arXiv:0904.2101 [math.NT], 2009.

%F a(n+1) = min{ m>a(n): A035263(m)=A035263(n+1) }.

%F a(n)=2n+1, if A007814(n) is even. a(n)=2n+2, if A007814(n) is odd.

%F A010060(a(n))=1-A010060(n)

%F For n>=1, A010060(a(n))= A010060(A004760(n+1)). See also A160230. [_Vladimir Shevelev_, May 05 2009]

%e n=2 is not in A003159. So a(2) is the smallest number larger than a(1)=3 which is not in A003159. This excludes 4 and 5 which are in A003159 and leads to a(2)=6.

%t a35263[n_] := 1 - Mod[IntegerExponent[n, 2], 2];

%t a[1] = 3; a[n_] := a[n] = For[k = a[n - 1] + 1, True, k++, If[a35263[k] == a35263[n], Return[k]]];

%t Array[a, 66] (* _Jean-François Alcover_, Jul 28 2018 *)

%o (PARI) is(n) = valuation(n, 2)%2==0; \\ A003159

%o nexta(a, n) = {my(k=a+1, isn = is(n)); while (is(k) != isn, k++); k;};

%o lista(nn) = {my(a = 3); print1(a, ", "); for (n=2, nn, a = nexta(a, n); print1(a, ", "););} \\ _Michel Marcus_, Dec 15 2018

%Y Cf. A003159, A007814, A010060, A160216, A159619.

%Y Cf. A004760, A160230. [_Vladimir Shevelev_, May 05 2009]

%K nonn

%O 1,1

%A _Vladimir Shevelev_, May 04 2009

%E Edited by _R. J. Mathar_, May 08 2009