A285786 -id:A285786 - cp's OEIS frontend

A285388 a(n) = numerator of ((1/n) * Sum_{k=0..n^2-1} binomial(2k,k)/4^k).

1, 35, 36465, 300540195, 79006629023595, 331884405207627584403, 22292910726608249789889125025, 11975573020964041433067793888190275875, 411646257111422564507234009694940786177843149765, 56592821660064550728377610673427602421565368547133335525825

Offset: 1

Ralf Steiner, Apr 18 2017

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

Indranil Ghosh, Table of n, a(n) for n = 1..40

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)).

[Numerator( n*(n^2+1)*Catalan(n^2)/2^(2*n^2-1) ): n in [1..21]]; // G. C. Greubel, Dec 11 2021

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 *)

A285388(n) = numerator((2^(1 - 2*(n^2)))*n*binomial(2*(n^2), n^2)); \\ Antti Karttunen, Apr 27 2017

a(n) = m=n*binomial(2*n^2, n^2);m>>valuation(m,2) \\ David A. Corneth, Apr 27 2017

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

[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

a(n) is numerator of n*binomial(2 n^2, n^2)/2^(2*n^2 - 1). - Ralf Steiner, Apr 26 2017
a(n) = numerator(n*A201555(n) / (A060757(n)/2)) = n*A201555(n) / 2^(A285717(n)) = A000265(n*A201555(n)). [Using Ralf Steiner's formula and A285717(n) <= A056220(n), cf. A285406.] - Antti Karttunen, Apr 27 2017
Limit_{i->oo} a(i)*A285389(i+1)/(a(i+1)*A285389(i)) = 1. - Ralf Steiner, May 03 2017

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
Formula section edited by M. F. Hasler, May 02 2017
Edited by N. J. A. Sloane, May 10 2017

A285738 Greatest prime less than 2*n^2 for n > 1, a(1) = 1.

Original entry on oeis.org

1, 7, 17, 31, 47, 71, 97, 127, 157, 199, 241, 283, 337, 389, 449, 509, 577, 647, 719, 797, 881, 967, 1051, 1151, 1249, 1327, 1453, 1567, 1669, 1789, 1913, 2039, 2161, 2311, 2447, 2591, 2731, 2887, 3041, 3191, 3361, 3527, 3697, 3863, 4049, 4231, 4409, 4603, 4801

Offset: 1

Ralf Steiner, Apr 25 2017

a(n) for n>1 is prime. Further the upper part of at least n in i well-ordered prime factors p_i(n) of the numerator of 2^(1-2 n^2) n binomial(2 n^2, n^2) (A285388(n)) consists of only single factors which form especially a complete part of the prime numbers p with 3 < 2(n-1)^2 < p <= a(n) < 2n^2. Thus the complete union of {2,3} and {p_i(m)} for m from 2 to n gives all prime numbers p <= a(n).

Alternative definitions are "Greatest prime factor of the numerator of 2^(1-2 n^2) n binomial(2 n^2, n^2)". and "Greatest prime factor of numerator of sum{k=0..n^2-1}(binomial(2k,k)/4^k)/n". - David A. Corneth, Apr 26 2017

Cf. A006530, A285388, A000040 (prime numbers), A285786 (Number of primes in interval).

Table[Last[FactorInteger[Numerator[2^(1-2 n^2) n Binomial[2 n^2, n^2]]][[All, 1]]], {n, 1, 30}]

a(n) = my(f = factor(sum(k = 0, n^2-1, (binomial(2*k, k)/4^k))/n)[, 1]); f[#f] \\ David A. Corneth, Apr 25 2017

a(n) = if(n==1,1,my(f=factor(n*binomial(2*n^2, n^2))[,1]); f[#f]) \\ David A. Corneth, Apr 26 2017

a(n) = if(n==1,return(1));my(i=2*n^2); while(!isprime(i), i--); i \\ David A. Corneth, Apr 26 2017

a(n) = A006530(A285388(n)).

a(31)-a(49) from David A. Corneth, Apr 25 2017

New name from David A. Corneth, Apr 26 2017

A285269 Number of (odd) primes between 2n^2 and 2(n+1)^2.

Original entry on oeis.org

0, 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

Offset: 0

M. F. Hasler, May 02 2017

nonn

Essentially the same as A285786, except for the offset and initial values.

			a(0) = 0 because between 2*0^2 = 0 and 2*1^2 = 2, there are no (odd) primes.
a(1) = 3 because between 2*1^2 = 2 and 2*2^2 = 8, there are the 3 (odd) primes 3, 5 and 7.
a(2) = 3 because between 2*2^2 = 8 and 2*3^2 = 18, there are the 3 primes 11, 13 and 17.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Join[{0}, Differences[PrimePi[2*Range[100]^2]]] (* Paolo Xausa, Nov 22 2024 *)

a(n)=primepi(2*(n+1)^2-1)-primepi(2*n^2)

a(n) = A285786(n+1), for all n >= 2.

First differences of A278114, a(n) = A278114(n+1) - A278114(n), for n > 0.

A289279 Number of odd composite numbers in ]2(n-1)^2, 2n^2[.

Original entry on oeis.org

0, 0, 2, 3, 5, 6, 8, 9, 11, 10, 14, 15, 18, 18, 19, 21, 24, 23, 27, 28, 28, 32, 31, 34, 35, 38, 39, 39, 41, 44, 46, 47, 48, 49, 50, 57, 51, 56, 59, 63, 59, 65, 61, 67, 67, 69, 73, 72, 73, 77, 78, 82, 80, 80, 85, 84, 87, 90, 90, 94, 98, 90, 97, 102, 100

Offset: 1

Ralf Steiner, Jul 01 2017

nonn

Cf. A285786, A006880.

Table[Count[Select[Range[2(n-1)^2,2 n^2],!PrimeQ[#]&&OddQ[#]&&(#>1) &], _Integer],{n,1,100}]
Table[2 n - 1 - (PrimePi[2 n^2] - PrimePi[2 (n - 1)^2]), {n, 1, 100}] (* Ralf Steiner, Jul 30 2017 *)

a(n) = sum(k=2*(n-1)^2, 2*n^2, (k%2) && (k!=1) && !isprime(k)); \\ Michel Marcus, Jul 01 2017

a(n) = 2n - 1 - A285786(n). - Ralf Steiner, Jul 30 2017

cp's OEIS Frontend

A285388 a(n) = numerator of ((1/n) * Sum_{k=0..n^2-1} binomial(2k,k)/4^k).

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A285738 Greatest prime less than 2*n^2 for n > 1, a(1) = 1.

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A285269 Number of (odd) primes between 2*n^2 and 2*(n+1)^2.

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A289279 Number of odd composite numbers in ]2*(n-1)^2, 2*n^2[.

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A285269 Number of (odd) primes between 2n^2 and 2(n+1)^2.

A289279 Number of odd composite numbers in ]2(n-1)^2, 2n^2[.