A092553 -id:A092553 - cp's OEIS frontend

A367709 Decimal expansion of BesselJ(0,2*sqrt(2)) (negated).

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

Offset: 0

Ilya Gutkovskiy, Nov 28 2023

			-0.19654809527046820004079337208793223...

Cf. A091681, A092553, A367574, A367712.

RealDigits[BesselJ[0, 2 Sqrt[2]], 10, 100][[1]]

Equals Sum_{k>=0} (-2)^k / k!^2.

A383216 Primes p which are preceded and followed by gaps whose difference is greater than 2*log(p).

Original entry on oeis.org

113, 127, 523, 887, 907, 1087, 1129, 1151, 1277, 1327, 1361, 1669, 1693, 1931, 1951, 1973, 2203, 2311, 2333, 2477, 2557, 2971, 2999, 3163, 3251, 3299, 3469, 4049, 4297, 4327, 4523, 4547, 4783, 4861, 5119, 5147, 5237, 5351, 5381, 5531, 5557, 5591, 5749, 5779, 5981

Offset: 1

Alain Rocchelli, Apr 19 2025

nonn

Primes prime(k) such that abs(prime(k-1)-2*prime(k)+prime(k+1)) > 2*log(prime(k)), where log is the natural logarithm.

			113 is a term because abs(109-2*113+127)=12 and 2*log(113)=9.4548.
127 is a term because abs(113-2*127+131)=10 and 2*log(127)=9.6884.

Cf. A036263, A383215, A092553.

Select[Prime[Range[2,782]],Abs[NextPrime[#,-1]-2#+NextPrime[#]]>2Log[#]&] (* James C. McMahon, Apr 27 2025 *)

forprime(P=3, 6000, my(M=P-precprime(P-1), Q=nextprime(P+1)-P, AR1=min(M,Q), AR2=max(M,Q), AR0=2*log(P)); if(AR2-AR1>AR0, print1(P,", ")));

Limit_{n->oo} n / PrimePi(a(n)) = 1/e^2 (A092553).

A322506 Factorial expansion of 1/exp(2) = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

0, 0, 0, 3, 1, 1, 3, 0, 6, 4, 7, 5, 2, 9, 9, 8, 10, 8, 9, 1, 13, 18, 1, 2, 8, 15, 26, 10, 22, 1, 18, 9, 20, 10, 2, 6, 13, 19, 16, 38, 38, 3, 32, 5, 39, 24, 7, 27, 14, 41, 20, 39, 32, 7, 20, 35, 44, 50, 24, 34, 51, 14, 39, 47, 49, 15, 61, 54, 60, 52, 34, 60, 32, 72, 48, 12, 67, 52, 22, 48

Offset: 1

G. C. Greubel, Dec 12 2018

nonn

			1/exp(2) = 0 + 0/2! + 0/3! + 3/4! + 1/5! + 1/6! + 3/7! + 0/8! + 6/9! +...

Index entries for factorial base representation

Cf. A092553 (decimal expansion), 0 U A001204 (continued fraction).

Cf. A054977 (e), A067840 (e^2), A068453 (sqrt(e)), A237420 (1/e).

SetDefaultRealField(RealField(250));  [Floor(Exp(-2))] cat [Floor(Factorial(n)*Exp(-2)) - n*Floor(Factorial((n-1))*Exp(-2)) : n in [2..80]];

With[{b = 1/E^2}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]]

default(realprecision, 250); b = exp(-2); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", "))

b=exp(-2);
def a(n):
    if (n==1): return floor(b)
    else: return expand(floor(factorial(n)*b) -n*floor(factorial(n-1)*b))
[a(n) for n in (1..80)]

A366134 Number of primes between prime(n) and prime(n)+2*log(prime(n)), exclusive.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 3, 2, 2, 1, 0, 1, 2, 1, 0, 2, 1, 2, 2, 1, 2, 1, 1, 3, 2, 1, 0, 0, 3, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 0, 3, 2, 1, 0, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 3, 2, 2, 1, 2, 3, 2, 1, 1, 2, 2, 2, 2

Offset: 1

Alain Rocchelli, Sep 30 2023

nonn

Inspired by A365573.

Cf. A365573, A354841, A092553.

a[n_]:=PrimePi[Prime[n]+2Log[Prime[n]]]-PrimePi[Prime[n]]; Array[a,95] (* Stefano Spezia, Sep 30 2023 *)

a(n) = primepi(prime(n)+2*log(prime(n))) - primepi(prime(n))

Conjecture: Limit_{N->oo} (Ratio_{n=1..N} a(n)=0) = 1/e^2 (A092553).

cp's OEIS Frontend

A367709 Decimal expansion of BesselJ(0,2*sqrt(2)) (negated).

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A383216 Primes p which are preceded and followed by gaps whose difference is greater than 2*log(p).

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A322506 Factorial expansion of 1/exp(2) = Sum_{n>=1} a(n)/n!.

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A366134 Number of primes between prime(n) and prime(n)+2*log(prime(n)), exclusive.

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