A057705 -id:A057705 - cp's OEIS frontend

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

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

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)}.

A135505 a(0) = 1; a(n) = [product_(i = 1..n) prime(i)^i] - 1, where prime(i) is i-th prime.

Original entry on oeis.org

1, 1, 17, 2249, 5402249, 870037764749, 4199506113235182749, 1723219765760312626547490749, 29266411525287522788837599332989370749, 52713275010243038997421106186697438702252144407249, 22176856087751973465466098269669474342964368337745368642450857249

Offset: 0

Ctibor O. Zizka, Feb 19 2008

Gheorghe Coserea, Table of n, a(n) for n = 0..50
C. K. Caldwell and Y. Gallot, On the primality of n!-1 and 2*3*..*p -1, Math. Comp., Volume 71, Number 237, Pages 441-448.
Leo Corry, Number crunching vs. number theory: computers and FLT, from Kummer to SWAC (1850-1960) and beyond, Archive for History of Exact Sciences, Vol. 62, No. 4 (July 2008), pp. 393-455.
Roland Queme, Some applications of Kummer and Stickelberger relations, arXiv:math/0601136 [math.NT], 2006.

Cf. A057588, A002110, A057705.

a(n) = { if (n <= 0, return(1)); prod(i = 1, n, prime(i)^i) - 1;}
vector(11, i, a(i-1))  \\ Gheorghe Coserea, Aug 24 2015

a(n) = A076954(n)-1, n>0. - R. J. Mathar, Nov 01 2009

Converted references to links - R. J. Mathar, Oct 30 2009

A135516 a(0)=1; a(n) = (Product_{i=1..n} prime(i)^2) - 1, where prime(i) is the i-th prime.

Original entry on oeis.org

1, 3, 35, 899, 44099, 5336099, 901800899, 260620460099, 94083986096099, 49770428644836899, 41856930490307832899, 40224510201185827416899, 55067354465423397733736099, 92568222856376731590410384099

Offset: 0

Ctibor O. Zizka, Feb 19 2008

Sequence can be generalized: a(0)=1; a(n) = (Product_{i=1..n} prime(i)^r) - 1, where prime(i) is the i-th prime.

G. C. Greubel, Table of n, a(n) for n = 0..99 [Offset shifted by _Georg Fischer_, Jun 18 2021]
A. Adelberg, S. Hong and W. Ren, Bounds on divided universal Bernoulli numbers and universal Kummer congruences, Proc. Amer. Math. Soc., Volume 136, Number 1, 2008, Pages 61-71,
Alexei A. Panchishkin, Generalized Kummer congruences and p-adic families of motives, arXiv:math/9503218 [math.NT], 1995.

Cf. A057588, A057705, A002110.

A002110 := proc(n) mul(ithprime(i),i=1..n) ; end:
A135516 := proc(n) if n =0 then 1; else (A002110(n)+1)*(A002110(n)-1) ; fi ; end: seq(A135516(n),n=0..20) ; # R. J. Mathar, Feb 28 2008

Join[{1},Rest[#-1&/@FoldList[Times,1,Prime[Range[15]]^2]]] (* Harvey P. Dale, Oct 02 2011 *)
Join[{1}, Table[Product[Prime[i]^(2), {i,1,n}] - 1, {n,1,15}]] (* G. C. Greubel, Oct 17 2016 *)

a(n) = if(n==0, 1, prod(k=1, n, prime(k)^2) - 1); \\ Michel Marcus, Oct 17 2016

a(n) = A061742(n-1)-1 = (A002110(n)+1)*(A002110(n)-1) for n>1. - R. J. Mathar, Feb 28 2008

Offset corrected by Georg Fischer, Jun 18 2021

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

A228486 Near primorial primes: primes p such that p+1 or p-1 is a primorial number (A002110).

Original entry on oeis.org

2, 3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309

Offset: 1

Jonathan Vos Post, Aug 22 2013

nonn

The next few terms are too large to add: 317#-1, 337#-1, 379#+1, 991#-1. - Charles R Greathouse IV, Sep 12 2013

Physicsforums, comment defining this sequence.

Cf. A000040, A002110, A057705, A228485.

A057705 UNION {primes p such that p-1 is a primorial number} = {primes p such that p+1 is a primorial number (A002110)} UNION {primes p such that p-1 is a primorial number}.

a(1)-a(2), a(7), a(11) inserted by Charles R Greathouse IV, Sep 12 2013

A286424 Number of partitions of p_n# into parts (q, k) both coprime to p_n#, with q prime and k nonprime, where p_n# = A002110(n).

Original entry on oeis.org

0, 0, 1, 1, 4, 110, 1432, 23338, 397661, 8193828, 212858328, 5941706227

Offset: 0

Michael De Vlieger, May 08 2017

Number of totative pairs (q, k) such that prime q + k nonprime = p_n# and both gcd(q, p_n#) = 1 and gcd(k, p_n#) = 1, with p_n < q <= pi(p_n#), where pi(p_n#) = A000849(n) - n = A048862(n).
Primes p_n < q <= pi(p_n#) are greater than the greatest prime factor of p_n# = p_n, and are thus coprime to p_n#. By the definition of primorial, we need not consider p >= p_n, as these p are divisors of p_n#, i.e., gcd(p, p_n#) = p. Since the totatives of m can be paired such that a + b = m, we need only determine if (p_n# - q) is not prime in order to count pairs (q, k).
a(n) < floor(A005867(n)/2).
a(n) <= A048862(n).
The totative pair (q,1) = (p_n# - 1, 1) is counted by a(n) for n in A057704, with (p_n# - 1) appearing in A057705.

			a(0) = 0 by definition. A002110(0) = 1; 1 is coprime to all numbers; the only possible totative pair is (1,1) and this does not include both a prime and a nonprime.
a(1) = 0 since, of the floor(A005867(1)/2) = 1 totative pair (1,1) of A002110(1) = 2, none include a both a prime and a nonprime.
a(2) = 1 since, the only totative pair (1,5) of A002110(1) = 6 includes both a prime and a nonprime.
a(3) = 1 since only (1,29) includes both a prime and a nonprime.
a(4) = 4 since (23,187), (41,169), (67,143), (89,121) include a both a prime and a nonprime.

C. K. Caldwell, The Prime Glossary, Primorial.
Eric Weisstein's World of Mathematics, Totative.

Cf. A000849, A002110, A005867, A048862, A048863, A057704, A057705, A116979, A117929.

Table[Function[P, Count[Prime@ Range[n + 1, PrimePi[P]], q_ /; ! PrimeQ[P - q]]]@ Product[Prime@ i, {i, n}], {n, 0, 9}] (* Michael De Vlieger, May 08 2017 *)

a(n) = (A000010(A002110(n)) - A048863(n)) - 2*A117929(A002110(n))
     = (A005867(n) - A048863(n)) - 2*A117929(A002110(n))
     = A048862(n) - 2*A117929(A002110(n)).

a(11) from Giovanni Resta, May 09 2017

cp's OEIS Frontend

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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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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A135505 a(0) = 1; a(n) = [product_(i = 1..n) prime(i)^i] - 1, where prime(i) is i-th prime.

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A135516 a(0)=1; a(n) = (Product_{i=1..n} prime(i)^2) - 1, where prime(i) is the i-th prime.

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

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A248585 Decimal expansion of the value of the continued fraction constructed from prime primorials minus 1.

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