A026491 -id:A026491 - cp's OEIS frontend

A159619 Slowest increasing sequence beginning with 4 such that n and a(n) are either both evil or both odious.

4, 7, 9, 11, 12, 15, 16, 19, 20, 23, 25, 27, 28, 31, 33, 35, 36, 39, 41, 43, 44, 47, 48, 51, 52, 55, 57, 59, 60, 63, 64, 67, 68, 71, 73, 75, 76, 79, 80, 83, 84, 87, 89, 91, 92, 95, 97, 99, 100, 103, 105, 107, 108, 111, 112, 115, 116, 119, 121, 123, 124, 127, 129, 131, 132, 135, 137

Offset: 1

Vladimir Shevelev, Apr 17 2009, Apr 27 2009, May 04 2009

(i) Theorem: For every initial value a(1) > 4, a minimum index n exists such that the a(n) obtained from that initial value coincides with this sequence here. Thus there exist essentially two slowest increasing sequences with this type of evil/odious congruence: A159615 and this one here.
(ii) In connection with this theorem, one can generalize to slowest increasing sequences a_m(n), a_m(1)=m, which let n and a(n) be at the same time in or not in some increasing sequence c(n). (This sequence here is c = A000069, m=4.)
We define a rank r of c as the minimum value a_r(1) such that for sufficiently large n (n depending on m) all sequences a_m(n), m>r, coincide with a_r(n).
In particular, c(n)=A004760(n+1) has rank r=2, and A000069 has rank r=3.
The problems are: 1) to find a sequence of rank r >= 4; 2) to find the rank of primes or to prove that it does not exist (in case of which it could be defined as infinity).
There is a conjecture arising in Sequence Machine that a(n) = A026491(2+n)-1. This appears to be true: Here we start from on odious or evil number and apply a minimum number of van-Eck-Transforms (of A171898) to reach a value larger than a(n-1). The Dekking formula in A026491 says that A026491 is essentially a partial sum of the backward van-Eck-Transforms, and in a (vague) manner this seems to match.
- R. J. Mathar, Jun 24 2021

Amiram Eldar, Table of n, a(n) for n = 1..10000
Hsien-Kuei Hwang, S. Janson and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016.
Hsien-Kuei Hwang, S. Janson and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.
Jon Maiga, Computer-generated formulas for A159619, Sequence Machine.
Vladimir Shevelev, Several results on sequences which are similar to the positive integers, arXiv:0904.2101 [math.NT], 2009.

Cf. A000069, A001969, A159615, A007814, A004760, A159559, A159560.

read("transforms") ; isA000069 := proc(n) option remember ; RETURN( type(wt(n), 'odd') ) ; end:
A159619 := proc(n) option remember; if n = 1 then 4; else for a from procname(n-1)+1 do if isA000069(a) = isA000069(n) then RETURN(a) ; fi; od: fi; end:
seq(A159619(n), n=1..120) ; # R. J. Mathar, Mar 25 2010

a[n_] := 2 * n + If[EvenQ[n] || EvenQ[IntegerExponent[n+1, 2]], 3, 2]; Array[a, 100] (* Amiram Eldar, Aug 30 2024 *)

a(n) = 2 * n + if(!(n % 2) || !(valuation(n+1, 2) % 2), 3, 2); \\ Amiram Eldar, Aug 30 2024

a(n) = 2n+3 if n*A007814(n+1) is even, and a(n) = 2n+2 otherwise.

Edited and extended by R. J. Mathar, Mar 25 2010

A171900 Backwards van Eck transform of A010060.

Original entry on oeis.org

0, 0, 1, 3, 2, 2, 1, 3, 1, 3, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 1, 3, 1, 3, 2, 2, 1, 3, 1, 3, 1, 3, 2, 2, 1, 3, 1, 3, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 1, 3, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 1, 3, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 3, 1, 3, 1, 3, 2, 2, 1, 3, 1, 3, 1, 3, 2, 2, 1, 3, 1

Offset: 1

N. J. A. Sloane, Oct 22 2010

nonn

See A171898 for definition. This assumes the offset of A010060 is taken to be 1.
A161916 gives the forwards van Eck transform of A010060.
Since A001285(n) = 1+A010060(n) differ only by a constant, this is also the Backwards van Eck Transform of A001285. - R. J. Mathar, Jun 24 2021

Cf. A010060, A161916, A171898. See also A026491.

A026491(n-1)-A026491(n-2) = a(n) for n>=3. - Michel Dekking, Apr 11 2019

cp's OEIS Frontend

A159619 Slowest increasing sequence beginning with 4 such that n and a(n) are either both evil or both odious.

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