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A127977
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The minimum excess in the prime race of odious primes versus evil primes in the interval (2^(n-1),2^n).
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1
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0, 1, 4, 7, 13, 19, 39, 53, 104, 138, 251, 334, 590, 715, 1353, 1855, 3659, 5221, 10484, 14933, 27491, 35474, 68816, 97342, 186405, 265255
(list; graph; refs; listen; history; internal format)
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OFFSET
| 5,3
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COMMENTS
| Shevelev conjectures (p.2) that for all natural numbers n other than 5 and 6, the number of evil primes not exceeding n <= the number of odious primes not exceeding n. Odious primes are A027697. Evil primes are A027699.
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LINKS
| Vladimir Shevelev, A Conjecture on Primes and a Step towards Justification, Jun 06 2007, table 1, p. 2, arXiv:0706.0786
Vladimir Shevelev, On excess of the odious primes (arXiv:0707.1761v2 [math.NT])
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EXAMPLE
| OdiPrimePi(x) for x >= 32 is 6, 6, 6, 6, 6, 7, 7, 7, 7, 8,.. and EviPrimePi(x) for x>=32 is 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6,...
The difference OdiPrimePi(x)-EviPrimePi(x) is 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3,.. so the minimum of the difference in the interval 2^(6-1)..2^6 is 1, yielding a(6)=1.
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MAPLE
| read("transforms") ; # see oeis.org/transforms.txt
isA000069 := proc(n) type(wt(n), 'odd') ; end proc;
isA027697 := proc(n) isprime(n) and isA000069(n) ; end proc:
isA027699 := proc(n) isprime(n) and not isA000069(n) ; end proc:
odiPi := proc(n) option remember; if n = 0 then 0; else an1 := procname(n-1) ; if isA027697(n) then an1+1 ; else an1 ; end if; end if; end proc:
eviPi := proc(n) option remember; if n = 0 then 0; else an1 := procname(n-1) ; if isA027699(n) then an1+1 ; else an1 ; end if; end if; end proc:
oddPi := proc(n) odiPi(n)-eviPi(n) ; end proc:
A127977 := proc(n) local a, x ; a := 2^(n+1) ; for x from 2^(n-1)+1 to 2^n-1 do a := min(a, oddPi(x)) ; end do: a; end proc:
for n from 5 do print(n, A127977(n)) ; end do; # R. J. Mathar, Sep 03 2011
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CROSSREFS
| Cf. A000069, A001969, A027697, A027699, A095005, A095006.
Sequence in context: A144730 A058570 A134781 * A100848 A051458 A068940
Adjacent sequences: A127974 A127975 A127976 * A127978 A127979 A127980
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KEYWORD
| nonn
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AUTHOR
| Jonathan Vos Post (jvospost3(AT)gmail.com), Jun 07 2007
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EXTENSIONS
| Published numbers corrected and checked up to n=23 by R. J. Mathar, Sep 03 2011
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