A088256 -id:A088256 - cp's OEIS frontend

A088257 Primorials P which have at least one prime neighbor; i.e., P+1 or P-1 or both are primes.

Original entry on oeis.org

1, 2, 6, 30, 210, 2310, 30030, 200560490130, 304250263527210, 23768741896345550770650537601358310

Offset: 1

Amarnath Murthy, Sep 27 2003

nonn

A supersequence of A088256.

Next term a(11) is 131 digits and is too large to include.

Alois P. Heinz, Table of n, a(n) for n = 1..19

Cf. A002110, A088256.

Select[FoldList[Times,1,Prime[Range[70]]],PrimeQ[#-1]||PrimeQ[#+1]&] (* Harvey P. Dale, Oct 22 2011 *)

More terms from Ray Chandler, Sep 28 2003

A088258 Factorials F which have at least one prime neighbor, i.e., F+1 or F-1 or both are primes.

Original entry on oeis.org

1, 2, 6, 24, 720, 5040, 39916800, 479001600, 87178291200, 10888869450418352160768000000, 265252859812191058636308480000000, 263130836933693530167218012160000000

Offset: 1

Amarnath Murthy, Sep 27 2003

nonn

Conjecture: sequence is infinite. But there are finitely many members which are sandwiched between twin primes.

			2 is in the sequence because 2 + 1 is prime.
6 is in the sequence because both 6 - 1 and 6 + 1 are prime.
24 is in the sequence because 24 - 1 is prime.

Robert Israel, Table of n, a(n) for n = 1..28

Cf. A000142, A088256, A088257, A088412.

select(t -> isprime(t+1) or isprime(t-1), [seq(n!,n=1..100)]); # Robert Israel, Aug 25 2016

Select[Range[32]!, Or @@ PrimeQ@ {# - 1, # + 1} &] (* Michael De Vlieger, Aug 25 2016 *)

a(n) = A088412(n)! = A000142(A088412(n)). - Robert Israel, Aug 25 2016

More terms from Ray Chandler, Sep 28 2003

A088259 Perfect powers which have at least one prime neighbor.

Original entry on oeis.org

1, 4, 8, 16, 32, 36, 100, 128, 196, 256, 400, 576, 676, 1296, 1600, 2916, 3136, 4356, 5476, 7056, 8100, 8192, 8836, 12100, 13456, 14400, 15376, 15876, 16900, 17956, 21316, 22500, 24336, 25600, 28900, 30976, 32400, 33856, 41616, 42436, 44100

Offset: 1

Amarnath Murthy, Sep 27 2003

nonn

If K is a term and K-1 is the neighboring prime then it must be a Mersenne prime.

Conjecture: sequence is infinite.

Harvey P. Dale, Table of n, a(n) for n = 1..322 (all terms up to 10 million)

Intersection of A001597 and A045718.

Cf. A088256, A088257, A088258.

Join[{1},Select[Range[45000],GCD@@FactorInteger[#][[All,2]]>1 && AnyTrue[ #+{1,-1},PrimeQ]&]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 05 2020 *)

Corrected and extended by Ray Chandler, Sep 28 2003

Offset changed by Andrew Howroyd, Sep 22 2024

A382785 a(n) is the least multiple of the n-th primorial such that both a(n)-1 and a(n)+1 are prime and the prime factors of a(n) do not exceed prime(n).

Original entry on oeis.org

4, 6, 30, 420, 2310, 180180, 4084080, 106696590, 892371480, 103515091680, 4412330782860, 29682952539240, 22514519501013540, 313986271960080720, 22750921955774182170, 912496437361321252440, 26918644902158976946980, 1290172194953476680815970, 1901713815361424627522739780

Offset: 1

Rory Pulvino, Apr 04 2025

nonn

a(n) is the smallest multiple k of the n-th primorial, prime(n)#, such that both k-1 and k+1 are prime and the prime factors of m = k/prime(n)# do not exceed prime(n).
From Michael S. Branicky, Apr 19 2025: (Start)
a(n) first differs from A060255(n) + 1 at n = 29.
a(349) has 1001 digits. (End)

			For a(2), (2*3)*1 = 6 and the first twin primes are 5, 7.
For a(3), (2*3*5)*1 = 30 and the first twin primes are 29, 31.
For a(4), (2*3*5*7)*2 = 420, the first twin primes are 419, 421 and 2 <= prime(4).
For a(5), (2*3*5*7*11)*1 = 2310 and the first twin primes are 3209, 3211.
For a(6), (2*3*5*7*11*13)*2*3 = 180180. the first twin primes are 180179, 180181 and 2, 3 <= prime(6).

Michael S. Branicky, Table of n, a(n) for n = 1..348

Supersequence of A088256.

Cf. A000040, A002110, A060255, A060256, A088257.

a[n_] := Module[{P,k},P=Product[Prime[i],{i, 1, n}];k = 1; While[!(PrimeQ[k*P-1] && PrimeQ[k*P+1]), k++];k*P] (* James C. McMahon, May 09 2025 *)

isok(k, p) = if (k>1, vecmax(factor(k)[,1])<=p, 1);
a(n) = my(P=vecprod(primes(n)), k=1, p=prime(n)); while(!(isok(k, p) && ispseudoprime(k*P-1) && ispseudoprime(k*P+1)), k++); k*P; \\ Michel Marcus, Apr 27 2025

from itertools import count
from sympy import factorint, isprime, prime, primorial
def a(n):
    pn, prn = prime(n), primorial(n)
    return next(k for m in count(1) if max(factorint(m), default=1)<=pn and isprime((k:=m*prn)-1) and isprime(k+1))
print([a(n) for n in range(1, 20)]) # Michael S. Branicky, Apr 18 2025

Data corrected by Michael S. Branicky, Apr 18 2025

A268480 Integers k such that A002110(k) is the average of two consecutive primes.

Original entry on oeis.org

2, 3, 5, 8, 38, 40, 64, 73, 89, 236, 480, 486

Offset: 1

Altug Alkan, Mar 21 2016

In other words, the primorial numbers that are considered are those of the form (p + q)/2 where p and q are consecutive primes. Note that the initial values of (p - q)/2 are 1, 1, 1, 23, 239, 191, 331, 373, 1021.
A088256 is a subsequence of these primorials, which in turn are a subsequence of A024675.
Numbers k such that A038711(k) = A060270(k). - Amiram Eldar, May 19 2024

			5 is a term because 2*3*5*7*11 = 2310 = (2309 + 2311)/2.
8 is a term because 2*3*5*7*11*13*17*19 = 9699690 = (9699667 + 9699713)/2.

Index entries for sequences related to primorial numbers.

Cf. A002110, A024675, A038711, A060270, A088256.

P:= 2: count:= 0:
for n from 2 to 500 do
  P:= P*ithprime(n);
  # first try d=1
  if isprime(P+1) then
    good:= isprime(P-1);
  elif isprime(P-1) then good:= false
  else
    for d from ithprime(n+1) by 2 do
      if igcd(d,P) > 1 then next fi;
      if isprime(P+d) then
        good:= isprime(P-d); break
      elif isprime(P-d) then
        good:= false; break
      fi
    od;
  fi;
  if good then
     count:= count+1;
     A[count]:= n;
  fi
od:
seq(A[i],i=1..count);  # Robert Israel, Aug 29 2016

prim[n_] := Times @@ Prime[Range[n]]; Select[Range[2, 100], Total[NextPrime[(p = prim[#]), {-1, 1}]] == 2*p &] (* Amiram Eldar, May 19 2024 *)

a002110(n) = prod(k=1, n, prime(k));
for(n=2, 1e3, if((nextprime(a002110(n)) - a002110(n)) == (a002110(n) - precprime(a002110(n))), print1(n, ", ")))

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A088257 Primorials P which have at least one prime neighbor; i.e., P+1 or P-1 or both are primes.

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A088258 Factorials F which have at least one prime neighbor, i.e., F+1 or F-1 or both are primes.

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A088259 Perfect powers which have at least one prime neighbor.

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A382785 a(n) is the least multiple of the n-th primorial such that both a(n)-1 and a(n)+1 are prime and the prime factors of a(n) do not exceed prime(n).

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A268480 Integers k such that A002110(k) is the average of two consecutive primes.

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