cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A054588 Length of gap between consecutive primes containing n!.

Original entry on oeis.org

2, 6, 14, 8, 12, 54, 30, 22, 14, 30, 90, 20, 90, 76, 90, 78, 190, 60, 62, 104, 186, 204, 190, 96, 44, 168, 254, 108, 188, 80, 38, 290, 174, 258, 98, 44, 170, 136, 132, 176, 180, 156, 292, 190, 312, 156, 142, 158, 450, 120, 130, 350, 132, 610, 384, 392, 430, 410
Offset: 3

Views

Author

Jeff Burch, Apr 14 2000

Keywords

Crossrefs

Cf. A058054. - Ivan Neretin, Jul 29 2015

A340013 The prime gap, divided by two, which surrounds n!.

Original entry on oeis.org

1, 3, 7, 4, 6, 27, 15, 11, 7, 15, 45, 10, 45, 38, 45, 39, 95, 30, 31, 52, 93, 102, 95, 48, 22, 84, 127, 54, 94, 40, 19, 145, 87, 129, 49, 22, 85, 68, 66, 88, 90, 78, 146, 95, 156, 78, 71, 79, 225, 60, 65, 175, 66, 305, 192, 196, 215, 205, 420, 101, 186, 213, 160
Offset: 3

Views

Author

Robert G. Wilson v, Jan 09 2021

Keywords

Comments

A theorem states that between (n+1)! + 2 and (n+1)! + (n+1) inclusive, there are n consecutive composite integers, namely 2, 3, 4, ..., n, n+1.
Records: 1, 3, 7, 27, 45, 95, 102, 127, 145, 146, 156, 225, 305, 420, 804, 844, 1173, 1671, 1725, 1827, 2570, 2930, 3318, 5142, 5946, 6837, 7007, 8208, 10221, ..., .

Examples

			For a(1), there are no positive primes which surround 1!. Therefore a(1) is undefined.
For a(2), there are two contiguous primes {2, 3} with 2 being 2!. The prime gap is 1. However, the two primes do not surround 2!, so a(2) is undefined.
For a(3), the following set of numbers, {5, 6, 7}, with 3! being in the middle. The prime gap is 2; therefore, a(3) = 1.
For a(4), the following set of numbers, {23, 24, 25, 26, 27, 28, 29} with 4! in between the two primes 23 & 29. The prime gap is 6, so a(4) = 3.

Links

Crossrefs

Cf. A000142, A002981, A006990, A033932, A033933, A037151, A054588, A058054, A073308.

Programs

  • Maple
    a:= n-> (f-> (nextprime(f-1)-prevprime(f+1))/2)(n!):
seq(a(n), n=3..70);  # Alois P. Heinz, Jan 09 2021
  • Mathematica
    a[n_] := (NextPrime[n!, 1] - NextPrime[n!, -1])/2; Array[a, 70, 3]
  • PARI
    a(n) = (nextprime(n!+1) - precprime(n!-1))/2; \\ Michel Marcus, Jan 11 2021
  • Python
    from sympy import factorial, nextprime, prevprime
def A340013(n):
    f = factorial(n)
    return (nextprime(f)-prevprime(f))//2 # Chai Wah Wu, Jan 23 2021

Formula

a(n) = (A037151(n) - A006990(n))/2 = (A033932(n) + A033933(n))/2.
a(n) = A054588(n)/2 = A058054(n)/2. - Alois P. Heinz, Jan 09 2021

A340802 Number of composite numbers between the largest noncomposite number <= n! and the smallest noncomposite number >= n!.

Original entry on oeis.org

0, 0, 1, 5, 13, 7, 11, 53, 29, 21, 13, 29, 89, 19, 89, 75, 89, 77, 189, 59, 61, 103, 185, 203, 189, 95, 43, 167, 253, 107, 187, 79, 37, 289, 173, 257, 97, 43, 169, 135, 131, 175, 179, 155, 291, 189, 311, 155, 141, 157, 449, 119, 129, 349, 131, 609, 383, 391, 429
Offset: 1

Views

Author

Alois P. Heinz, Jan 21 2021

Keywords

Examples

			a(4) = 5: 24, 25, 26, 27, 28.
a(5) = 13: 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126.
a(6) = 7: 720, 721, 722, 723, 724, 725, 726.

Links

Crossrefs

Cf. A000040, A002808, A008578, A046933, A054588, A058054, A340013.

Programs

  • Maple
    prevprime(2):= 1:
a:= n-> (f-> max(nextprime(f-1)-prevprime(f+1)-1, 0))(n!):
seq(a(n), n=1..64);
  • Mathematica
    a[n_] := If[n<3, 0, NextPrime[n!] - NextPrime[n!, -1] - 1];
Array[a, 100] (* Jean-François Alcover, Jan 29 2021 *)

Formula

a(n) = A058054(n)-1 for n >= 2.
a(n) = A054588(n)-1 for n >= 3.
a(n) = 2 * A340013 - 1 for n >= 3.

A182487 Nextprime(F(n)) - prevprime(F(n)), where F(n) is the n-th Fibonacci number.

Original entry on oeis.org

3, 4, 4, 6, 4, 6, 6, 14, 10, 10, 6, 6, 8, 18, 12, 24, 16, 10, 6, 12, 30, 12, 24, 42, 30, 24, 60, 24, 30, 34, 30, 36, 46, 12, 36, 18, 34, 24, 24, 30, 36, 52, 72, 16, 22, 48, 44, 50, 34, 20, 20, 28, 44, 50, 40, 92, 60, 86, 16, 52, 48, 66, 46, 168, 50, 174, 36
Offset: 4

Views

Author

Alex Ratushnyak, May 02 2012

Keywords

Comments

Smallest prime following Fibonacci(n) minus largest prime immediately preceding Fibonacci(n). Starting from Fibonacci(4), because for n<4 there is no prime preceding Fibonacci(n).

Examples

			a(0) = A014208(4) - A180422(0) = 5 - 2 = 3,
a(7) = A014208(11) - A180422(7) = 97-83 = 14.

Links

Crossrefs

Cf. A014208, A180422, A013633, A058043, A058054, A161503.
Cf. A079677 (distance from F(n) to the nearest prime).

Programs

  • Maple
    a:= n-> (f-> nextprime(f)-prevprime(f))(combinat[fibonacci](n)):
seq(a(n), n=4..100);  # Alois P. Heinz, Jul 29 2015
  • Mathematica
    Table[f = Fibonacci[n]; NextPrime[f] - NextPrime[f, -1], {n, 4, 100}] (* T. D. Noe, May 02 2012 *)

Formula

a(n) = A014208(n+4) - A180422(n).
Showing 1-4 of 4 results.

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