A341804 -id:A341804 - cp's OEIS frontend

A058233 Primes p such that p#+1 is divisible by the next prime after p.

2, 17, 1459, 2999

Offset: 1

Carlos Rivera, Dec 01 2000

No additional terms through the 100000th prime. - Harvey P. Dale, Mar 12 2014

a(5) > prime(1400000) = 22182343. - Robert Price, Apr 02 2018

			2*3*5*7*11*13*17+1 is divisible by 19.

Carlos Rivera, Puzzle 117: Certain p#+1 values, The Prime Puzzles and Problems Connection.

Cf. A006862, A066735, A341804.

primorial[n_] := Product[ Prime[k], {k, 1, PrimePi[n]}]; Select[ Prime[ Range[1000]], Divisible[ primorial[#] + 1, NextPrime[#]] &] (* Jean-François Alcover, Aug 19 2013 *)
Module[{prs=Prime[Range[500]]},Transpose[Select[Thread[{Rest[ FoldList[ Times, 1,prs]], prs}], Divisible[ First[#]+1, NextPrime[Last[#]]]&]][[2]]] (* Harvey P. Dale, Mar 12 2014 *)

from sympy import nextprime
A058233_list, p, q, r = [], 2, 3, 2
for _ in range(10**3):
    if (r+1) % q == 0:
        A058233_list.append(p)
    r *= q
    p, q = q, nextprime(q) # Chai Wah Wu, Sep 27 2021

A341805 Numbers k such that (product of first k primes)-1 is divisible by the (k+1)-th prime.

Original entry on oeis.org

0, 2, 4, 9823712

Offset: 1

Jeppe Stig Nielsen, Feb 20 2021

			4 is a member because 2*3*5*7-1 (product of first 4 primes, minus one) is divisible by the 5th prime, 11.
9823712 is a member because 2*3*5*...*176078267-1 is divisible by 176078293, where 176078267 is the 9823712th prime.

Cf. A079276, A081618, A338543, A341804, A341812.

isok(k) = ((vecprod(primes(k)) - 1) % prime(k+1)) == 0; \\ Michel Marcus, Mar 03 2021

a(n) = A000720(A341804(n)) - 1.

A341812 Primes p such that p#-1 is divisible by the next prime after p.

Original entry on oeis.org

3, 7, 176078267

Offset: 1

Jeppe Stig Nielsen, Feb 20 2021

Here p# means A034386(p).

The next prime after p is then in A341804.

			The prime 7 is a member of this sequence because 2*3*5*7-1=209 is divisible by 11, the next prime immediately following 7.
11 is NOT a member of this sequence because 2*3*5*7*11-1=2309 is not divisible by 13 (leaves a remainder of 8), the next prime after 11.

Carlos Rivera, Puzzle 117. Certain p#+1 values, contains message from Wilfred Whiteside mentioning this sequence.

Cf. A057588, A058233, A341804, A341805.

isok(p) = if (isprime(p), my(a=Mod(1,nextprime(1+p))); forprime(q=2,p,a*=q); (a == 1)); \\ Michel Marcus, Mar 03 2021

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A058233 Primes p such that p#+1 is divisible by the next prime after p.

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A341805 Numbers k such that (product of first k primes)-1 is divisible by the (k+1)-th prime.

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A341812 Primes p such that p#-1 is divisible by the next prime after p.

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Author

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Programs

PARI