A006794 -id:A006794 - cp's OEIS frontend

A248584 Decimal expansion of the value of the continued fraction constructed from prime primorials plus 1.

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

Offset: 0

Jean-François Alcover, Oct 09 2014

			0.318248165083690124777685589996787844788657122331533049467...

Marek Wolf, "Continued fractions constructed from prime numbers" arXiv:1003.4015 [math.NT] Sep 26 2010, p. 14.

Cf. A005234, A006794, A018239, A057705, A242072, A247856, A247858, A247860, A248585.

RealDigits[N[FromContinuedFraction[{0, 3, 7, 31, 211, 2311, 200560490131}], 102]] // First

A248585 Decimal expansion of the value of the continued fraction constructed from prime primorials minus 1.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Oct 09 2014

			0.198630157303503810875201233614346862875870630898479777625647...

Marek Wolf, Continued fractions constructed from prime numbers, arXiv:1003.4015 [math.NT], Sep 26 2010, p. 14.

Cf. A005234, A006794, A018239, A057705, A242072, A247856, A247858, A247860, A248584.

cf = {0, 5, 29, 2309, 30029, 304250263527209, 23768741896345550770650537601358309}; RealDigits[N[FromContinuedFraction[cf], 102]] // First

A250294 Primes p such that p#-1 is a semiprime, where # is the primorial (A034386).

Original entry on oeis.org

7, 17, 29, 31, 43, 59, 71, 73, 97, 101, 223, 233, 257, 439, 503, 709, 859, 863, 1013

Offset: 1

Eric Chen, Dec 24 2014

1153 and 1381 are also terms. - Amiram Eldar, Feb 16 2020

a(20) >= 1091. 1091# - 1 is a 458-digit composite with no known factors. - Hugo Pfoertner, Feb 05 2021

			a(2) = 17 so 17# - 1 = 510509 = 61 * 8369 is a semiprime.

FactorDB Factorization of 1013#-1.
FactorDB Status of 1091#-1.
Hisanori Mishima, Factorizations of many number sequences, PI Pn - 1 (n = 1 to 110) (2013).

Cf. A001221, A001358, A002110, A006794, A034386, A057588, A078781, A250293.

p = Select[Range[101], PrimeQ]; p[[ Position[FoldList[Times, p] - 1, ?(PrimeOmega[#] == 2 &)] //Flatten ]] (* _Amiram Eldar, Feb 16 2020 *)

A001221(A034386(a(n)) - 1) = 2. - Amiram Eldar, Feb 16 2020

a(15)-a(18) using factordb.com from Amiram Eldar, Feb 16 2020
a(19) using factordb.com from Hugo Pfoertner, Feb 05 2021
Edited by Max Alekseyev, Aug 26 2021

A333058 0, 1, or 2 primes at primorial(n) +- 1.

Original entry on oeis.org

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

Offset: 0

Frank Ellermann, Mar 06 2020

nonn

a(n) = 0 marks a prime gap size of at least 2*prime(n+1)-1, e.g., primorial(8) +- prime(9) = {9699667,9699713} are primes, gap 2*23-1.
Mathworld reports that it is not known if there are an infinite number of prime Euclid numbers.
The tables in Ondrejka's collection contain no further primorial twin primes after {2309,2311} = primorial(13) +- 1 up to primorial(15877) +- 1 with 6845 digits.

			a(2) = a(3) = a(5) = 2: 2*3 +-1 = {5,7}, 6*5 +-1 = {29,31} and 210*11 +-1 = {2309,2311} are twin primes.
a(1) = a(4) = a(6) = 1: 1, 30*7 - 1 = 209 and 2310*13 + 1 = 30031 are not primes.
a(7) = 0: 510509 = 61 * 8369 and 510511 = 19 * 26869 are not primes.

H. Dubner, A new primorial prime, J. Rec. Math., 21 (No. 4, 1989), 276.

Chris K. Caldwell, the top 20: Primorial, 2012.
H. Dubner & N. J. A. Sloane, Correspondence, 1991, on A005234.
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 30029.
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 9699667.
Rudolf Ondrejka, The Top Ten: a Catalogue of Primal Configurations, 2001, tables 20, 20A, 20B.
Eric Weisstein's World of Mathematics, Primorial Prime.
Eric Weisstein's World of Mathematics, Euclid Number.

Cf. A096831, A002110 (primorials, p#), A057706.
Cf. A006862 (Euclid, p#+1), A005234 (prime p#+1), A014545 (index prime p#+1).
Cf. A057588 (Kummer, p#-1), A006794 (prime p#-1), A057704 (index prime p#-1).
Cf. A010051, A088411 (where a(n) is positive), A088257.

p:= proc(n) option remember; `if`(n<1, 1, ithprime(n)*p(n-1)) end:
a:= n-> add(`if`(isprime(p(n)+i), 1, 0), i=[-1, 1]):
seq(a(n), n=0..120);  # Alois P. Heinz, Mar 18 2020

primorial[n_] := primorial[n] = Times @@ Prime[Range[n]];
a[n_] := Boole@PrimeQ[primorial[n] - 1] + Boole@PrimeQ[primorial[n] + 1];
a /@ Range[0, 105] (* Jean-François Alcover, Nov 30 2020 *)

S = ''                     ;  Q = 1
do N = 1 to 27
   Q = Q * PRIME( N )
   T = ISPRIME( Q - 1 ) + ISPRIME( Q + 1 )
   S = S || ',' T
end N
S = substr( S, 3 )
say S                      ;  return S

a(n) = [ isprime(primorial(n) - 1) ] + [ isprime(primorial(n) + 1) ].

a(n) = Sum_{i in {-1,1}} A010051(primorial(n) + i).

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A248584 Decimal expansion of the value of the continued fraction constructed from prime primorials plus 1.

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A248585 Decimal expansion of the value of the continued fraction constructed from prime primorials minus 1.

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A250294 Primes p such that p#-1 is a semiprime, where # is the primorial (A034386).

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A333058 0, 1, or 2 primes at primorial(n) +- 1.

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