A248785 -id:A248785 - cp's OEIS frontend

A248682 Decimal expansion of Sum_{n >= 0} (floor(n/2)!)^2/n!.

2, 9, 4, 5, 5, 9, 9, 4, 3, 4, 8, 7, 4, 8, 6, 0, 3, 1, 1, 6, 3, 9, 1, 8, 0, 6, 7, 3, 4, 5, 9, 6, 9, 3, 9, 8, 4, 2, 5, 2, 5, 0, 3, 3, 3, 1, 6, 3, 7, 9, 9, 1, 6, 2, 2, 7, 2, 8, 7, 8, 6, 6, 0, 9, 2, 3, 3, 8, 8, 7, 2, 7, 2, 1, 1, 2, 3, 1, 4, 5, 6, 3, 2, 7, 4, 7

Offset: 1

Clark Kimberling, Oct 11 2014

Limit_{x -> inf} Sum {n=0..inf} (Floor[n/x])!^x/n! = e (A001113).
For A248682: x = 2; A248683: x = 3; A248684: x = 4; A248685: x = 5. - Robert G. Wilson v, Feb 22 2016
Let n} denote the swinging factorial A056040(n), then the constant equals Sum_{n>=0} 1/n} and is sometimes called the swinging constant e}. ("e}" is written in TeX $e\wr$). For a proof that it equals 3^(1/2)*(2/3)^3*Pi + 4/3 see the link to Mathematics Stack Exchange. - Peter Luschny, Jul 22 2022

			2.94559943487486031163918067345969398425250...

Georg Fischer, Table of n, a(n) for n = 1..1000 (first 149 terms from Clark Kimberling)
Jan Eerland, Answer to question 3689772, Mathematics Stack Exchange, 2021. See also question 3692793.
Index entries for transcendental numbers

Cf. A001113, A248683, A248684, A248785, A248664, A056040 (swinging factorial).

RealDigits[Sum[(Floor[n/2])!^2/n!, {n, 0, 400}], 10, 111][[1]]
RealDigits[4/3+8Pi/Sqrt[243],10,111][[1]] (* Robert G. Wilson v, Feb 10 2016 *)

suminf(n=0, ((n\2)!)^2/n!) \\ Michel Marcus, Feb 11 2016

Equals Sum_{n >= 0} (n!^2)*p(2,n)/(2*n + 1)!, where p(k,n) is defined at A248664.
Equals Sum_{n >= 0} (floor(n/2)!)^2/n! = Sum_(n >= 1) (3n^2 - 7n + 6)/C(2n, n) = 4/3 + 8*Pi/sqrt(243). - Robert G. Wilson v, Feb 11 2016
Equals 1 + Integral_{x>=0} 1/(x^2 - x + 1)^2 dx. - Amiram Eldar, Nov 16 2021

A177833 Numbers k such that k^2 - 13 and k^2 + 13 are primes.

Original entry on oeis.org

4, 12, 18, 72, 84, 114, 198, 354, 378, 588, 612, 618, 864, 912, 948, 1032, 1068, 1134, 1320, 1410, 1428, 1452, 1500, 1830, 1956, 2046, 2058, 2172, 2298, 2448, 2634, 2748, 2844, 2856, 3192, 3246, 3390, 3474, 3846, 3906, 4092, 4182, 4506, 4842, 4884, 4890

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), May 14 2010

nonn

			4^2 - 13 = 3 = prime(2), 4^2 + 13 = 29 = prime(10).
12^2 - 13 = 131 = prime(32), 12^2 + 13 = 157 = prime(37).
948^2 - 13 = 898691 = prime(71194), 948^2 + 13 = prime(71195), first case that they are consecutive primes.

J. Matousek and J. Nesetril, Diskrete Mathematik: eine Entdeckungsreise, Springer-Lehrbuch, 2. Aufl., Berlin, 2007

Zak Seidov, Table of n, a(n) for n = 1..3000

Cf. A138375, A154648, A176969, A176978, A248785.

[n: n in [4..1000]| IsPrime(n^2-13) and IsPrime(n^2+13)]; // Vincenzo Librandi, Nov 30 2010

with(numtheory): A248785:=n->`if`(isprime(n^2-13) and isprime(n^2+13), n, NULL): seq(A248785(n), n=1..10^4); # Wesley Ivan Hurt, Oct 13 2014

Select[Range[2,5000,2],AllTrue[#^2+{13,-13},PrimeQ]&] (* Harvey P. Dale, May 28 2024 *)

More terms from Vincenzo Librandi, May 16 2010

Name edited by Michel Marcus, Nov 25 2024

A248790 Numbers n with the property that p = n^2 - 11 and q = n^2 + 11 are consecutive primes.

Original entry on oeis.org

510, 720, 1200, 2190, 4350, 4980, 5040, 5250, 5670, 6810, 8280, 8490, 9150, 10140, 10650, 11430, 12510, 13800, 13980, 14160, 14640, 14700, 14820, 15000, 15750, 16890, 17220, 18180, 18270, 18750, 19110, 20940, 21270, 22050, 24000, 24570, 24720, 24990, 25620, 25920, 26520

Offset: 1

Zak Seidov, Oct 14 2014

nonn

All terms are == 0 (mod 30).

			n=510, p=260089=prime(22845), q=260111=prime(22846).

Zak Seidov, Table of n, a(n) for n = 1..1000

Subsequence of A176683 and of A075190. E.g., a(1)=510=A075190(62)=A176683(6).

Cf. A248785.

Select[Range[30000],With[{c=#^2-11},PrimeQ[c]&&NextPrime[c]==c+22&]] (* Harvey P. Dale, Apr 03 2025 *)

isok(n) = isprime(p=n^2-11) && isprime(q=n^2+11) && (q==nextprime(p+1)); \\ Michel Marcus, Oct 14 2014

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A248682 Decimal expansion of Sum_{n >= 0} (floor(n/2)!)^2/n!.

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A177833 Numbers k such that k^2 - 13 and k^2 + 13 are primes.

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A248790 Numbers n with the property that p = n^2 - 11 and q = n^2 + 11 are consecutive primes.

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