A006794 -id:A006794 - cp's OEIS frontend

A066269 Indices of primes in A066268.

2, 3, 33

Offset: 1

Patrick De Geest, Dec 16 2001

The 33rd term, prime 16268...99999, contains 759 digits.
Next term is bigger than 150. - Stefan Steinerberger, May 06 2007
Next term is bigger than 450. - Michael S. Branicky, Nov 14 2024

Carlos Rivera, Puzzle 118. Primorial product numbers, The Prime Puzzles & Problems Connection.
Eric Weisstein's World of Mathematics, Primorial.

Cf. A066266, A066267, A066268, A002110, A006939, A006794.

Select[Range[65], PrimeQ[Product[Product[Prime[i],{i,1,n}],{n,1,#}]-1]&] (* Stefan Steinerberger, Apr 12 2006 *)

Offset changed from 0 to 1 by Harry J. Smith, Feb 08 2010

A057706 Smaller of twin primes whose average is a primorial number.

Original entry on oeis.org

5, 29, 2309

Offset: 1

Labos Elemer, Oct 24 2000

According to Caldwell, the next term, if it exists, has more than 100000 digits. - T. D. Noe, May 08 2012

			(5+7)/2 = 6 = 2*3, (29+31)/2 = 30 = 2*3*5, (2309+2311)/2 = 2310 = 2*3*5*7*11.

Chris K. Caldwell, Primorial prime.
Rudolf Ondrejka, The Top Ten: a Catalogue of Primal Configurations, 2001, tables 20A, 20B.
Eric Weisstein's World of Mathematics, Primorial Prime.
Index entries for sequences related to primorial numbers.

Cf. A000040 (primes), A002110 (primorials, p#).
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).

Select[FoldList[Times, Prime@ Range@ 40], AllTrue[# + {-1, 1}, PrimeQ] &] - 1 (* Michael De Vlieger, Jul 15 2017 *)

from sympy import isprime, prime, primerange
def auptoprimorial(limit):
  phash, alst = 1, []
  for p in primerange(1, prime(limit)+1):
    phash *= p
    if isprime(phash-1) and isprime(phash+1): alst.append(phash-1)
  return alst
print(auptoprimorial(5)) # Michael S. Branicky, May 29 2021

Offset corrected by Arkadiusz Wesolowski, May 08 2012

A066268 Product of first n primorials - 1.

Original entry on oeis.org

1, 11, 359, 75599, 174635999, 5244319079999, 2677277333530799999, 25968760179275365451999999, 5793445238736255798985527239999999, 37481813439427687898244906452608585199999999, 7517370874372838151564668004911177464757864075999999999

Offset: 1

Patrick De Geest, Dec 16 2001

nonn

			a(3) = (2)*(2*3)*(2*3*5) - 1 = 359.

Harry J. Smith, Table of n, a(n) for n = 1..37
Romeo Meštrović, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2023. - From N. J. A. Sloane, Jun 13 2012
Carlos Rivera, Puzzle 118. Primorial product numbers, The Prime Puzzles & Problems Connection.
Eric Weisstein's World of Mathematics, Primorial.

Cf. A066266, A066267, A066269, A002110, A006939, A006794.

Table[Times@@Table[Times@@Prime[Range[n]],{n,k}]-1,{k,40}]
(* or *)
pr2=1; Table[pr1=1; Do[pr1=pr1*Prime[n],{n,k}]; pr2=pr2*pr1; pr2-1,{k,40}] (* Jayanta Basu, May 12 2013 *)

a(n) = -1 + prod(k=1, n, prime(k)^(n-k+1)) \\ Andrew Howroyd, Dec 10 2024

a(n) = A006939(n) - 1. - Andrew Howroyd, Dec 10 2024

Offset changed from 0 to 1 by Harry J. Smith, Feb 08 2010

A060255 Smaller of twin primes {p, p+2} whose average p+1 = kq is the least multiple of the n-th primorial number q such that kq-1 and k*q+1 are twin primes.

Original entry on oeis.org

3, 5, 29, 419, 2309, 180179, 4084079, 106696589, 892371479, 103515091679, 4412330782859, 29682952539239, 22514519501013539, 313986271960080719, 22750921955774182169, 912496437361321252439, 26918644902158976946979, 1290172194953476680815969, 1901713815361424627522739779

Offset: 1

Labos Elemer, Mar 22 2001

nonn

a(349) has 1001 digits. - Michael S. Branicky, Apr 19 2025

			a(13) = -1 + (2*3*5*7*...*41)*k(13) = 304250263527210*74 and {22514519501013539, 22514519501013542} are the corresponding primes; k(13)=74 is the smallest suitable multiplier. Twin primes obtained from primorial numbers with k=1 multiplier seem to be much rarer (see A057706).
For j=1,2,3,4,5,6, a(j)=A001359(1), A059960(1), A060229(1), A060230(1), A060231(1), A060232(1) respectively.

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

Cf. A001359, A002110, A006794, A014545, A057706, A059960, A060229, A060230, A060231, A060232.

a(n) = {my(q = prod(k=1, n, prime(k))); for(k=1, oo, if (isprime(q*k-1) && isprime(q*k+1), return(q*k-1)););} \\ Michel Marcus, Jul 10 2018

from itertools import count
from sympy import primorial, isprime
def a(n):
    p = primorial(n)
    return next(m-1 for m in count(p, p) if isprime(m-1) and isprime(m+1))
print([a(n) for n in range(1, 20)]) # Michael S. Branicky, Apr 18 2025

a(n) = p = k(n)*q(n)-1, where q(n)=A002110(n) and k(n)=A060256(n) is the smallest integer whose multiplication by the n-th primorial yields p+1.

a(2)=5 corrected by Ray Chandler, Apr 03 2009

a(18) and beyond from Michael S. Branicky, Apr 18 2025

A103515 Primes of the form primorial P(k)*2^n-1 with minimal n, n>=0, k>=2.

Original entry on oeis.org

5, 29, 419, 2309, 30029, 1021019, 19399379, 892371479, 51757545839, 821495767572479, 14841476269619, 304250263527209, 54873078184468933509119, 2459559130353965639, 521426535635040715679, 15751252788463309939261439

Offset: 1

Lei Zhou, Feb 15 2005

nonn

Conjecture: sequence is defined for all k>=2

			P(2)*2^0-1=3*2-1=5 is prime, so a(2)=5;
P(4)*2^1-1=7*5*3*2*2-1=419 is prime, so a(4)=419;

Cf. A002110, A005234, A014545, A018239, A006794, A057704, A057705, A103153, A103154.

nmax = 2^2048; npd = 2; n = 2; npd = npd*Prime[n]; While[npd < nmax, tt = 1; cp = npd*tt - 1; While[ ! (PrimeQ[cp]), tt = tt*2; cp = npd*tt - 1]; Print[cp]; n = n + 1; npd = npd*Prime[n]]

A103782 a(n) = minimal m >= 0 that makes primorial P(n)*2^m-1 prime.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 1, 2, 3, 12, 1, 0, 22, 2, 4, 13, 12, 6, 1, 4, 1, 4, 0, 2, 9, 5, 6, 2, 1, 9, 17, 22, 7, 19, 73, 23, 12, 5, 27, 33, 64, 33, 5, 7, 41, 44, 35, 29, 3, 19, 6, 26, 5, 11, 9, 33, 34, 16, 63, 46, 8, 4, 24, 48, 0, 11, 0, 26, 6, 25, 17, 31, 6, 46, 33, 46, 17, 8, 61, 12, 23, 76, 20, 17

Offset: 2

Lei Zhou, Feb 15 2005

nonn

The values of n in A103515

			P(2)*2^0-1=5 is prime, so a(2)=0; P(9)*2^2-1=892371479 is prime, so a(9)=2;

Cf. A002110, A005234, A014545, A018239, A006794, A057704, A057705, A103513, A103515.

nmax = 2^2048; npd = 2; n = 2; npd = npd*Prime[n]; While[npd < nmax, tn = 0; tt = 1; cp = npd*tt - 1; While[(cp > 1) && (! (PrimeQ[cp])), tn = tn + 1; tt = tt*2; cp = npd*tt - 1]; Print[tn]; n = n + 1; npd = npd*Prime[n]]

A104876 Semiprimes of the form primorial(k) - 1.

Original entry on oeis.org

209, 510509, 6469693229, 200560490129, 13082761331670029, 1922760350154212639069, 557940830126698960967415389, 40729680599249024150621323469, 2305567963945518424753102147331756069, 232862364358497360900063316880507363069

Offset: 1

Jonathan Vos Post, Mar 28 2005

			4# - 1 = 209 = 11 * 19.
7# - 1 = 510509 = 61 * 8369.
10# - 1 = 6469693229 = 79 * 81894851.

Sebastian Martin Ruiz, A Result on Prime Numbers, Math. Gaz. 81, 269, 1997.
Eric Weisstein's World of Mathematics, Primorial.
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A001358, A002110, A034386, A005234, A014545, A018239, A006794, A057704, A057705, A104877.

Bigomega[n_]:=Plus@@Last/@FactorInteger[n]; SemiprimeQ[n_]:=Bigomega[n]==2; Primorial[n_]:=Product[Prime[i], {i, n}]; Select[Table[Primorial[n]-1, {n, 30}], SemiprimeQ] (* Ray Chandler, Mar 28 2005 *)

n# - 1 iff semiprime. Equals {A002110(i) - 1} intersection {A001358(j)}.

Entry revised by N. J. A. Sloane, Apr 01 2006

A068489 m for which prime(m) is the least prime dividing #prime(n) - 1, i.e., one less than primorial n-th prime (A057588).

Original entry on oeis.org

3, 10, 5, 343, 3248, 18, 16, 12, 22, 20324, 50, 9414916809095, 13120, 43, 8481, 1200361259, 196, 38, 10326732314, 65, 38, 34

Offset: 2

Lekraj Beedassy, Mar 11 2002

Since #P13 - 1 is a prime, see A006794, we need the number of primes less than or equal to #P13 - 1. The sequence continues, for n=14 to 23: 13120, 43, 8481, 1200361259, 196, 38, 10326732314, 65, 38, 34.

a(24) = pi(23768741896345550770650537601358309). - Donovan Johnson, Dec 08 2009

Romeo Meštrović, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2023. - From N. J. A. Sloane, Jun 13 2012
Hisanori Mishima, Factorization results #Pn (Primorial) - 1.

Cf. A000720, A006794, A057588, A057713, A068488.

Do[ Print[ PrimePi[ FactorInteger[ Product[ Prime[k], {k, 1, n}] - 1] [[1, 1]]]], {n, 2, 22} ]

a(n) = A000720(A057713(n)).

Edited and extended by Robert G. Wilson v, Mar 12 2002

a(13) from Donovan Johnson, Dec 08 2009

A103513 Primes of the form primorial(P(k))/2-2^n with minimal n, n>=0, k>=2.

Original entry on oeis.org

2, 13, 103, 1153, 15013, 255253, 4849843, 111546433, 3234846607, 100280245063, 3710369067401, 152125131763603, 6541380632280583, 307444891294245701, 16294579238595022363, 961380175077106319471, 58644190679703485491571

Offset: 1

Lei Zhou, Feb 15 2005

nonn

The Mathematica Program does not produce a(2). Conjecture: sequence is defined for all k>=2.

			P(2)/2=3, 3-2^0=2 is prime, so a(2)=2;
P(5)/2=1155, 1155-2^1=1153 is prime, so a(5)=1153;

Cf. A002110, A005234, A014545, A018239, A006794, A057704, A057705, A103154.

nmax = 2^8192; npd = 1; n = 2; npd = npd*Prime[n]; While[npd < nmax, tt = 2; cp = npd - tt; While[(cp > 1) && (! (PrimeQ[cp])), tt = tt*2; cp = npd - tt]; If[cp < 2, Print["*"], Print[cp]]; n = n + 1; npd = npd*Prime[n]]

A104877 Semiprimes of the form primorial(k) + 1.

Original entry on oeis.org

30031, 9699691, 223092871, 13082761331670031, 117288381359406970983271, 7858321551080267055879091, 40729680599249024150621323471, 267064515689275851355624017992791

Offset: 1

Jonathan Vos Post, Mar 28 2005

			6# + 1 = 2*3*5*7*11*13 + 1 = 30031 = 59 x 509.
8# + 1 = 2*3*5*7*11*13*17*19 + 1 = 9699691 = 347 x 27953.
9# + 1 = 2*3*5*7*11*13*17*19*23 + 1 = 223092871 = 317 x 703763.
14# + 1 = 2*3*5*7*11*13*17*19*23*29*31*37*41*43 + 1 = 13082761331670031 = 167 x 78339888213593.

Sebastian Martin Ruiz, A Result on Prime Numbers, Math. Gaz. 81, 269, 1997.
Eric Weisstein's World of Mathematics, Primorial.
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A001358, A002110, A034386, A005234, A014545, A018239, A006794, A057704, A057705, A104876.

Bigomega[n_]:=Plus@@Last/@FactorInteger[n]; SemiprimeQ[n_]:=Bigomega[n]==2; Primorial[n_]:=Product[Prime[i], {i, n}]; Select[Table[Primorial[n]+1, {n, 30}], SemiprimeQ] (* Ray Chandler, Mar 28 2005 *)
Select[FoldList[Times,Prime[Range[30]]]+1,PrimeOmega[#]==2&] (* Harvey P. Dale, Oct 13 2022 *)

n# + 1 iff semiprime. Equals {A002110(i) + 1} intersection {A001358(j)}.

cp's OEIS Frontend

A066269 Indices of primes in A066268.

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A057706 Smaller of twin primes whose average is a primorial number.

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A066268 Product of first n primorials - 1.

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A060255 Smaller of twin primes {p, p+2} whose average p+1 = k*q is the least multiple of the n-th primorial number q such that k*q-1 and k*q+1 are twin primes.

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A103515 Primes of the form primorial P(k)*2^n-1 with minimal n, n>=0, k>=2.

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Mathematica

A103782 a(n) = minimal m >= 0 that makes primorial P(n)*2^m-1 prime.

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Mathematica

A104876 Semiprimes of the form primorial(k) - 1.

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Mathematica

Formula

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A068489 m for which prime(m) is the least prime dividing #prime(n) - 1, i.e., one less than primorial n-th prime (A057588).

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A060255 Smaller of twin primes {p, p+2} whose average p+1 = kq is the least multiple of the n-th primorial number q such that kq-1 and k*q+1 are twin primes.