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A159619
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Slowest increasing sequence beginning with 4 such that n and a(n) are either both evil or both odious.
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10
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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; internal format)
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OFFSET
| 1,1
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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 is could be defined as infinity). [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Apr 23 2009]
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LINKS
| V. Shevelev, Several results on sequences which are similar to the positive integers, arXiv:0904.2102
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FORMULA
| a(n)=2n+3 if n*A007814(n+1) is even, and a(n)=2n+2 otherwise.
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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
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CROSSREFS
| Cf. A000069, A001969, A159615, A007814, A004760, A159559, A159560
Sequence in context: A010454 A053169 A007656 * A207017 A174724 A131827
Adjacent sequences: A159616 A159617 A159618 * A159620 A159621 A159622
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KEYWORD
| nonn,easy
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AUTHOR
| Vladimir Shevelev (shevelev(AT)bgu.ac.il), Apr 17 2009, Apr 27 2009, May 04 2009
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EXTENSIONS
| Edited and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 25 2010
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