A005234 -id:A005234 - cp's OEIS frontend

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

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

A250293 Primes p such that p#+1 is a semiprime, where # is the primorial (A034386).

Original entry on oeis.org

13, 19, 23, 43, 61, 67, 73, 83, 101, 139, 151, 173, 223, 251, 383, 457, 571, 673, 761

Offset: 1

Eric Chen, Dec 24 2014

The next candidate after 571 is 859. 859# + 1 is a 359-digit composite with no known factors. - Hugo Pfoertner, Feb 05 2021

			a(1) = 13 so 13# + 1 = 30031 = 59 * 509 is a semiprime.

factordb.com, Status of 859#+1.
Hisanori Mishima, Factorizations of many number sequences, PI Pn + 1 (n = 1 to 110) (2012).

Cf. A005234, A006862, A034386, A078778, A085725, A250294.

Prime[#]&/@(Module[{pmrl=FoldList[Times,Prime[Range[50]]]},Position[ pmrl, ?(PrimeOmega[ #+1]==2&)]]//Flatten) (* _Harvey P. Dale, Apr 29 2017 *)

a(n) = prime(A085725(n)). - Hugo Pfoertner, Feb 05 2021

a(16)-a(18) using factordb.com from Hugo Pfoertner, Feb 05 2021

Missing 571 inserted by Sean A. Irvine, Mar 03 2023

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)}.

A114432 Primes of the form 1 + product of the first k 4-almost primes A014613.

Original entry on oeis.org

17, 8126654054401

Offset: 1

Jonathan Vos Post, Feb 13 2006

nonn

The next term is too large to display here. - N. J. A. Sloane, Jul 30 2009
4-almost prime analog of primorial primes A005234 as indexed by A014545. In that sense, this sequence is indexed by (1, 8, ...). No more through product of first 16 of the 4-almost primes.
Terms are one more than the products of 4-almost primes up to 16, 81, 294, 513, 825, 1356, 1612, 2004, 2756, 7714, ... - Charles R Greathouse IV, Jul 28 2009

			a(1) = 17 because 1 + 16 = 1 + A014613(1) = 1 more than the first 4-almost prime is itself prime.
a(2) = 8126654054401 = 1 + (16 * 24 * 36 * 40 * 54 * 56 * 60 * 81) = 1 more than the product of the first 8 of the 4-almost primes and is prime.

Amiram Eldar, Table of n, a(n) for n = 1..8

Cf. A000040, A014613, A005234, A014545.

Select[FoldList[Times, Select[Range[200], PrimeOmega[#] == 4 &]] + 1, PrimeQ] (* Amiram Eldar, Jul 20 2025 *)

{1 + Product_{i=1..k} A014613(i)} INTERSECTION A000040.

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

Original entry on oeis.org

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

A114428 Primes of the form 1 + product of the first n semiprimes.

Original entry on oeis.org

5, 2161, 30241, 453601, 4495130640001, 152834441760001, 911300420785759804800001, 19660095637340203930960075575675174251117567173124497920000000001

Offset: 1

Jonathan Vos Post, Feb 13 2006

nonn

Semiprime 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 (1, 4, 5, 6, 11, 12, 39, ...).

The next term has 90 digits. - Harvey P. Dale, Sep 21 2011

			a(1) = 5 = 4 + 1 = 1 + A001358(1) = 1 + A112141(1) because 4 is the first semiprime and 5 is prime.
a(2) = 2161 because 2160 + 1 = 1 + A001358(1)*A001358(2)*A001358(3)*A001358(4) = 1 + A112141(4) = 1 + (4*6*9*10) is prime.
a(3) = 1 + A112141(5).
a(4) = 1 + A112141(6).
a(5) = 1 + A112141(11).
a(6) = 1 + A112141(12).
a(7) = (4 * 6 * 9 * 10 * 14 * 15 * 21 * 22 * 25 * 26 * 33 * 34 * 35 * 38 * 39 * 46 * 49 * 51 * 55 * 57 * 58 * 62 * 65 * 69 * 74 * 77 * 82 * 85 * 86 * 87 * 91 * 93 * 94* 95 * 106 * 111 * 115 * 118 * 119) + 1 is prime.

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

Select[#+1&/@FoldList[Times,1,Select[Range[200],PrimeOmega[#] == 2&]], PrimeQ] (* Harvey P. Dale, Sep 21 2011 *)

{a(n)} = {1 + A112141} INTERSECTION {A000040}.

a(7) added by Jonathan Vos Post, Dec 12 2010

cp's OEIS Frontend

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

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

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

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A250293 Primes p such that p#+1 is a semiprime, where # is the primorial (A034386).

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

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

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A114432 Primes of the form 1 + product of the first k 4-almost primes A014613.

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

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