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 A159619 Slowest increasing sequence beginning with 4 such that n and a(n) are either both evil or both odious. 10
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS (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). REFERENCES Hsien-Kuei Hwang, S Janson, TH 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; http://140.109.74.92/hk/wp-content/files/2016/12/aat-hhrr-1.pdf. Also 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 LINKS V. Shevelev, Several results on sequences which are similar to the positive integers, arXiv:0904.2101 [math.NT], 2009. FORMULA a(n)=2n+3 if n*A007814(n+1) is even, and a(n)=2n+2 otherwise. MAPLE 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 CROSSREFS Cf. A000069, A001969, A159615, A007814, A004760, A159559, A159560 Sequence in context: A273916 A053169 A007656 * A272454 A207017 A174724 Adjacent sequences:  A159616 A159617 A159618 * A159620 A159621 A159622 KEYWORD nonn,base,easy AUTHOR Vladimir Shevelev, Apr 17 2009, Apr 27 2009, May 04 2009 EXTENSIONS Edited and extended by R. J. Mathar, Mar 25 2010 STATUS approved

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Last modified December 6 14:15 EST 2019. Contains 329806 sequences. (Running on oeis4.)