A159615 The slowest increasing sequence beginning with a(1)=2 such that a(n) and n are both odious or both not odious.

2, 4, 5, 7, 9, 10, 11, 13, 15, 17, 19, 20, 21, 22, 23, 25, 27, 29, 31, 33, 35, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 99, 101, 103, 105, 107, 109, 111

Offset: 1

Author

Vladimir Shevelev, Apr 17 2009

Keywords

• nonn
• easy

Examples

			If n=3, then k=1, j=0, therefore a(6)=(10*3-4*0)/3=10.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Vladimir Shevelev, Several results on sequences which are similar to the positive integers, arXiv:0904.2101 [math.NT], 2009.

Crossrefs

Cf. A000069, A159559, A159560, A004760.

Programs

Maple

read("transforms") ; isA000069 := proc(n) option remember ; RETURN( type(wt(n),'odd') ) ; end:
A159615 := proc(n) option remember; if n = 1 then 2; else for a from procname(n-1)+1 do if isA000069(a) = isA000069(n) then RETURN(a) ; fi; od: fi; end:
seq(A159615(n),n=1..120) ; # R. J. Mathar, Aug 17 2009

Mathematica

odiousQ[n_] := OddQ[DigitCount[n, 2, 1]];
a[1] = 2; a[n_] := a[n] = For[k = a[n-1]+1, True, k++, If[FreeQ[Array[a, n-1], k] && odiousQ[n] && odiousQ[k] || !odiousQ[n] && !odiousQ[k], Return[k] ] ];
Array[a, 80] (* Jean-François Alcover, Dec 10 2017 *)

Formula

For n>=1, a(n)=min{m>a(n-1): A010060(m)=A010060(n)}.

a(2n+1)=2a(n)+1.

a(2n)=3n+1+j,if n=2^k+j; a(2n)=(10n-4j)/3,if n=2^k+2^(k-1)+j, where 0<=j<=2^(k-1)-1.

Extensions

Edited and extended by R. J. Mathar, Aug 17 2009