cp's OEIS Frontend

A160217 Minimal increasing sequence with a(1)=3 and the property that a(n) and n are both in or both not in A003159.

%I A160217 #15 Dec 15 2018 08:34:08
%S A160217 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 A160217 51,54,55,57,59,62,63,66,67,70,71,73,75,78,79,82,83,86,87,89,91,94,95,
%U A160217 97,99,102,103,105,107,110,111,114,115,118,119,121,123,126,127,129,131,134
%N A160217 Minimal increasing sequence with a(1)=3 and the property that a(n) and n are both in or both not in A003159.
%C A160217 The primes in this sequence give A160216.
%C A160217 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 A160217 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 A160217 a(n+1) = min{ m>a(n): A035263(m)=A035263(n+1) }.
%F A160217 a(n)=2n+1, if A007814(n) is even. a(n)=2n+2, if A007814(n) is odd.
%F A160217 A010060(a(n))=1-A010060(n)
%F A160217 For n>=1, A010060(a(n))= A010060(A004760(n+1)). See also A160230. [_Vladimir Shevelev_, May 05 2009]
%e A160217 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 A160217 a35263[n_] := 1 - Mod[IntegerExponent[n, 2], 2];
%t A160217 a[1] = 3; a[n_] := a[n] = For[k = a[n - 1] + 1, True, k++, If[a35263[k] == a35263[n], Return[k]]];
%t A160217 Array[a, 66] (* _Jean-François Alcover_, Jul 28 2018 *)
%o A160217 (PARI) is(n) = valuation(n, 2)%2==0; \\ A003159
%o A160217 nexta(a, n) = {my(k=a+1, isn = is(n)); while (is(k) != isn, k++); k;};
%o A160217 lista(nn) = {my(a = 3); print1(a, ", "); for (n=2, nn, a = nexta(a, n); print1(a, ", "););} \\ _Michel Marcus_, Dec 15 2018
%Y A160217 Cf. A003159, A007814, A010060, A160216, A159619.
%Y A160217 Cf. A004760, A160230. [_Vladimir Shevelev_, May 05 2009]
%K A160217 nonn
%O A160217 1,1
%A A160217 _Vladimir Shevelev_, May 04 2009
%E A160217 Edited by _R. J. Mathar_, May 08 2009