A160217 - OEIS

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

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OFFSET

1,1

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.

LINKS

Table of n, a(n) for n=1..66.

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

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]

EXAMPLE

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.

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 *)

PROG

(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

CROSSREFS

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

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

Sequence in context: A289182 A188974 A047558 * A228014 A188971 A176065