Search: a285738 -id:a285738
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A285388
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a(n) = numerator of ((1/n) * Sum_{k=0..n^2-1} binomial(2k,k)/4^k).
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+10
14
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1, 35, 36465, 300540195, 79006629023595, 331884405207627584403, 22292910726608249789889125025, 11975573020964041433067793888190275875, 411646257111422564507234009694940786177843149765, 56592821660064550728377610673427602421565368547133335525825
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graph;
refs;
listen;
history;
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OFFSET
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1,2
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COMMENTS
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Editorial comment: This sequence arose from Ralf Steiner's attempt to prove Legendre's conjecture that there is a prime between N^2 and (N+1)^2 for all N. - N. J. A. Sloane, May 01 2017
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LINKS
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FORMULA
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a(n) is numerator of n*binomial(2 n^2, n^2)/2^(2*n^2 - 1). - Ralf Steiner, Apr 26 2017
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MATHEMATICA
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Table[Numerator[Sum[Binomial[2k, k]/4^k, {k, 0, n^2-1}]/n], {n, 1, 10}]
Numerator[Table[2^(1-2 n^2) n Binomial[2 n^2, n^2], {n, 1, 10}]] (* Ralf Steiner, Apr 22 2017 *)
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PROG
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(PARI) a(n) = m=n*binomial(2*n^2, n^2); m>>valuation(m, 2) \\ David A. Corneth, Apr 27 2017
(Python)
from sympy import binomial, Integer
def a(n): return (Integer(2)**(1 - 2*n**2)*n*binomial(2*n**2, n**2)).numerator() # Indranil Ghosh, Apr 27 2017
(Magma) [Numerator( n*(n^2+1)*Catalan(n^2)/2^(2*n^2-1) ): n in [1..21]]; // G. C. Greubel, Dec 11 2021
(Sage) [numerator( n*(n^2+1)*catalan_number(n^2)/2^(2*n^2-1) ) for n in (1..20)] # G. C. Greubel, Dec 11 2021
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CROSSREFS
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Cf. A000079, A000265, A056220, A060757, A201555, A285389 (denominators), A285406, A280655 (similar), A190732 (2/sqrt(Pi)), A285738 (greatest prime factor), A285717, A285730, A285786, A286264, A000290 (n^2), A056220 (2*n^2 -1), A286127 (sum a(n-1)/a(n)).
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KEYWORD
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nonn,frac
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AUTHOR
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EXTENSIONS
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Edited (including the removal of the author's claim that this leads to a proof of the Legendre conjecture) by N. J. A. Sloane, May 01 2017
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STATUS
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approved
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A285786
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Number of primes p with 2(n-1)^2 < p <= 2n^2.
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+10
5
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1, 3, 3, 4, 4, 5, 5, 6, 6, 9, 7, 8, 7, 9, 10, 10, 9, 12, 10, 11, 13, 11, 14, 13, 14, 13, 14, 16, 16, 15, 15, 16, 17, 18, 19, 14, 22, 19, 18, 16, 22, 18, 24, 20, 22, 22, 20, 23, 24, 22, 23, 21, 25, 27, 24, 27, 26, 25, 27, 25, 23, 33, 28, 25, 29, 28, 31, 30, 33, 29
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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The author of the sequence conjectures that a(n) >= 1 for all n. This conjecture is similar to the famous conjecture made by Adrien-Marie Legendre that there is always a prime between n^2 and (n+1)^2, see A014085. - Antti Karttunen, May 01 2017
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LINKS
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FORMULA
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For all n except n=2, a(n) <= n.
(End)
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EXAMPLE
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For n = 1, the primes from 2*((1-1)^2) to 2*(1^2) (in semiopen range ]0, 2]) are: 2, thus a(1) = 1.
For n = 2, the primes from 2*((2-1)^2) to 2*(2^2) (in semiopen range ]2, 8]) are: 3, 5 and 7, thus a(2) = 3.
For n = 3, the primes from 2*((3-1)^2) to 2*(3^2) (in semiopen range ]8, 18]) are: 11, 13 and 17, thus a(3) = 3.
For n = 4, the primes from 2*((4-1)^2) to 2*(4^2) (in semiopen range ]18, 32]) are: 19, 23, 29 and 31, thus a(4) = 4.
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MAPLE
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R:= [0, seq(numtheory:-pi(2*n^2), n=1..100)]:
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MATHEMATICA
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Table[Length[Select[FactorInteger[Numerator[Table[2^(1 - 2 n^2) n Binomial[2 n^2, n^2], {n, 1, k}]]][[k]][[All, 1]], # > 2 (k - 1)^2 &]], {k, 1, 60}]
Flatten[{1, 2, Table[PrimePi[2 k^2] - PrimePi[2 (k - 1)^2], {k, 3, 60}]}]
(* Second program: *)
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PROG
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(Python)
from sympy import primepi
def a(n): return primepi(2*n**2) - primepi(2*(n - 1)**2) # Indranil Ghosh, May 01 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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