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

3, 6, 7, 9, 11, 14, 15, 18, 19, 22, 23, 25, 27, 30, 31, 33, 35, 38, 39, 41, 43, 46, 47, 50, 51, 54, 55, 57, 59, 62, 63, 66, 67, 70, 71, 73, 75, 78, 79, 82, 83, 86, 87, 89, 91, 94, 95, 97, 99, 102, 103, 105, 107, 110, 111, 114, 115, 118, 119, 121, 123, 126, 127, 129, 131, 134

Offset: 1

Author

Vladimir Shevelev, May 04 2009

Keywords

• nonn

Comments

The primes in this sequence give A160216.

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.

Examples

			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.
		

Links

• V. Shevelev, Several results on sequences which are similar to the positive integers, arXiv:0904.2101 [math.NT], 2009.

Crossrefs

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

Cf. A004760, A160230. [Vladimir Shevelev, May 05 2009]

Programs

Mathematica

a35263[n_] := 1 - Mod[IntegerExponent[n, 2], 2];
a[1] = 3; a[n_] := a[n] = For[k = a[n - 1] + 1, True, k++, If[a35263[k] == a35263[n], Return[k]]];
Array[a, 66] (* Jean-François Alcover, Jul 28 2018 *)

PARI

is(n) = valuation(n, 2)%2==0; \\ A003159
nexta(a, n) = {my(k=a+1, isn = is(n)); while (is(k) != isn, k++); k;};
lista(nn) = {my(a = 3); print1(a, ", "); for (n=2, nn, a = nexta(a, n); print1(a, ", "););} \\ Michel Marcus, Dec 15 2018

Formula

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

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

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

For n>=1, A010060(a(n))= A010060(A004760(n+1)). See also A160230. [Vladimir Shevelev, May 05 2009]

Extensions

Edited by R. J. Mathar, May 08 2009