A014545 -id:A014545 - cp's OEIS frontend

A114430 Primes of the form 1 + product of the first n 3-almost primes A014612.

97, 32920473601, 1448500838401, 65182537728001, 1491301685600774317670400000001, 48235157779343672198731287466250036763794299837586774072944798728192000000000000000001

Offset: 1

Jonathan Vos Post, Feb 13 2006

3-almost prime analog of primorial primes A005234 (primes p such that 1 + product of primes up to p is prime) as indexed by A014545 (n such that n-th Euclid number (A006862(n)) = 1 + (Product of first n primes) is prime). In that sense, this sequence is indexed by (2, 8, 9, 10, 19, ...).

			a(1) = 97 = 96 + 1 = 1 + (8 * 12) = 1 + A014612(1)*A014612(2) = 1 more than the product of the first 2 of the 3-almost primes and is prime.
a(2) = 32920473601 = 1 + (8 * 12 * 18 * 20 * 27 * 28 * 30 * 42) = 1 more than the product of the first 8 of the 3-almost primes and is prime.
a(3) = 1 more than the product of the first 9 of the 3-almost primes and is prime.
a(4) = 1 more than the product of the first 10 of the 3-almost primes and is prime.
a(5) = 1 more than the product of the first 19 of the 3-almost primes and is prime.

Cf. A000040, A001358, A014612, A002110, A005234, A014545, A112141.

Select[Rest[FoldList[Times,1,Select[Range[250],PrimeOmega[#]==3&]]]+1,PrimeQ] (* Harvey P. Dale, Dec 21 2013 *)

{a(n)} = {1 + Prod[from i = 1 to n] A014612(i)} INTERSECTION {A000040}.

One more term (a(6)) from Harvey P. Dale, Dec 21 2013

A216205 Incidences of n such that A006862(n) - n! is prime where A006862 are the Euclid numbers.

Original entry on oeis.org

0, 1, 2, 6, 8, 19, 94, 226, 2277, 2742, 2868

Offset: 0

Frank M Jackson, Mar 12 2013

			a(3) = 6 because A006862(6) - 6! = 30031-720 = 29311 and is the 3rd such prime.

Cf. A006862, A014545.

primeproduct[q_] := Product[Prime[r], {r, 1, q}]; nextterm[n_] := (p=n+1; While[!PrimeQ[primeproduct[p]+1-p!], p++]; p); Table[Nest[nextterm, 0, m], {m, 1, 5}] (* changing 5 to 10 will give all 10 terms but takes a long time *)

A264855 Integers n such that A002110(n)^2 - A002110(n) + 1 is prime.

Original entry on oeis.org

1, 2, 4, 5, 10, 14, 15, 20, 23, 46, 96, 281, 367, 542, 1380, 1395

Offset: 1

Altug Alkan, Nov 26 2015

Initial primes of the form A002110(n)^2 - A002110(n) + 1 are 3, 31 and 43891.

Intersection of this sequence and A014545 gives the values of n such that A002110(n)^3 + 1 is semiprime.

			a(1) = 1 because 2^2 - 2 + 1 = 3 is prime.
a(2) = 2 because 6^2 - 6 + 1 = 31 is prime.
a(3) = 4 because 210^2 - 210 + 1 = 43891 is prime.

Cf. A002110, A014545, A092061.

Select[Range@ 400, PrimeQ[#^2 - # + 1 &@ Product[Prime@ k, {k, #}]] &] (* Michael De Vlieger, Nov 28 2015 *)

a002110(n) = prod(k=1, n, prime(k));
for(n=0, 1e3, if(ispseudoprime(a002110(n)^2 - a002110(n) + 1), print1(n, ", ")))

A305400 a(n) = round(1/(A073918(n)/prime(n)# - 1)), where A073918(n) = min { prime p | omega(p-1) = n } and p# = product of primes <= p.

Original entry on oeis.org

1, 2, 6, 30, 210, 2310, 2, 1, 3, 3, 14, 200560490130, 2, 4, 2, 8, 7, 2, 2, 2, 4, 9, 7, 3, 2, 5, 7, 4, 13, 27, 2, 3, 3, 10, 3, 8, 9, 4, 41, 7, 4, 5, 7, 32, 5, 32, 6, 5, 7, 11, 7, 4, 5, 13, 5, 21, 10, 19, 27, 8, 7, 3, 6, 51, 15, 10, 10, 15, 8, 21, 17, 29

Offset: 0

M. F. Hasler, May 31 2018

nonn

We conjecture that lim inf A073918(n)/A002110(n) = 1 but the value of the lim sup is unknown. Therefore we consider x defined as A073918(n)/A002110(n) = 1 + 1/x, and a(n) = round(x).

We have lim sup a(n) = oo <=> lim inf A073918(n)/A002110(n) = 1, and lim inf a(n) = m <=> (2m + 1)/(2m - 1) >= lim sup A073918(n)/A002110(n) >= (2m + 3)/(2m + 1), where the first inequality only holds for m >= 1.

			For 0 <= n <= 5,  A073918(n) = prime(n)# + 1, therefore a(n) = prime(n)#.
For n = 6, the smallest prime p such that p - 1 has 6 distinct prime factors is prime(5)#*prime(8) + 1, therefore a(n) = round(prime(6)/(prime(8) + 1/prime(5)# - prime(6))) = 2.

Cf. A073918, A002110, A014545, A305398, A305399.

apply( a(n)=1\/(A073918(n)/factorback(primes(n))-1), [0..99])

a(n) = round(A002110(n)/(A073918(n) - A002110(n))).

a(n) = A002110(n) <=> n in A014545 <=> primorial(n) + 1 is prime.

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).

cp's OEIS Frontend

A114430 Primes of the form 1 + product of the first n 3-almost primes A014612.

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A216205 Incidences of n such that A006862(n) - n! is prime where A006862 are the Euclid numbers.

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A264855 Integers n such that A002110(n)^2 - A002110(n) + 1 is prime.

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A305400 a(n) = round(1/(A073918(n)/prime(n)# - 1)), where A073918(n) = min { prime p | omega(p-1) = n } and p# = product of primes <= p.

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

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