cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A136349 Numbers k of the form Product_{j=1..m} prime(j) such that k-1 is prime.

Original entry on oeis.org

6, 30, 2310, 30030, 304250263527210, 23768741896345550770650537601358310
Offset: 1

Views

Author

Enoch Haga, Dec 25 2007

Keywords

Comments

This sequence is different from A121069 and A002110.
Compute the product of k consecutive sequences of prime factors 2,3,5,7, etc. where k=1,2,3,4,5, etc. When N is preceded by prime N-1 add N to the sequence.
a(7) = 1 9361386640 7008231634 7142505431 2320082662 8976125715 6376190696 2414215012 3698566371 7909694733 5243680669 6075314756 2914824028 4399976570 - copied from Data field by Michael B. Porter, Mar 30 2013
Next term (a(8)) is too large to be included: see A006794. - M. F. Hasler, May 02 2008
The next 7 terms in the sequence are a(7) = p# 2..89 (shown in full above), a(8) = p# 2..317, a(9) = p# 2..337, a(10) = p# 2..991, a(11) = p# 2..1873, a(12) = p# 2..2053, a(13) = p# 2..2377, where p# indicates a primorial. - Jeff Hall, Apr 05 2021

Examples

			a(3)=30 where the prime factors are 2,3,5; since N-1=29, prime, N=30 is added to the sequence.

Crossrefs

Cf. A136350, A136351, A136352, A002110, A121069.

Programs

  • Mathematica
    Select[FoldList[Times,1,Prime[Range[70]]],PrimeQ[#-1]&]  (* Harvey P. Dale, Jan 09 2011 *)
  • PARI
    c=0;t=1;vector(7,n,until( ispseudoprime( -1+t*=prime(c++)),);t)

Formula

a(n) = A057705(n) + 1 = A034386( A006794(n) ). - M. F. Hasler, May 02 2008

Extensions

Edited by M. F. Hasler, May 02 2008, May 30 2008

A136352 Primorials P for which neither P-1 nor P+1 is prime.

Original entry on oeis.org

510510, 9699690, 223092870, 6469693230, 7420738134810, 13082761331670030, 614889782588491410, 32589158477190044730, 1922760350154212639070, 117288381359406970983270, 7858321551080267055879090
Offset: 1

Views

Author

Enoch Haga, Dec 25 2007

Keywords

Comments

This sequence is different from A121069 and A002110.
For almost all primes p, p# is in this sequence by Brun's theorem. - Charles R Greathouse IV, Sep 14 2015

Examples

			13# = 30030 is preceded by a prime but is not followed by a prime. 17# = 510510 is neither preceded nor followed by a prime.

Links

Crossrefs

Cf. A136349, A136350, A136351, A002110, A121069.

Programs

  • Mathematica
    Select[FoldList[Times, 1, Prime[Range[19]]],!PrimeQ[#+1]&&!PrimeQ[#-1]&] (* James C. McMahon, May 08 2025 *)

Formula

Compute P = product of the first k primes. If P is neither preceded nor followed by a prime add P to the sequence.

Extensions

Edited by and more terms from Charles R Greathouse IV, Sep 29 2008

A136350 Primorial numbers #p such that (#p)-1 is composite.

Original entry on oeis.org

210, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 13082761331670030, 614889782588491410, 32589158477190044730, 1922760350154212639070, 117288381359406970983270
Offset: 1

Views

Author

Enoch Haga, Dec 25 2007

Keywords

Comments

This sequence is different from A121069 and A002110.

Examples

			a(1)=210 because 209=A002110(4)-1 is not prime.

Crossrefs

Cf. A136349, A136351, A136352, A002110, A121069.

Programs

  • Mathematica
    Select[FoldList[Times, 1, Prime[Range[18]]],CompositeQ[#-1]&] (* James C. McMahon, May 08 2025 *)

Formula

{A002110(j): A002110(j)-1 in A002808}. - R. J. Mathar, Jul 23 2008

Extensions

Edited and extended by R. J. Mathar, Jul 23 2008

A136353 First odd composite N divisible by precisely the first n odd primes with N-2 prime.

Original entry on oeis.org

9, 15, 105, 1155, 15015, 255255, 4849845, 111546435, 9704539845, 100280245065, 18551845337025, 152125131763605, 98120709987525225, 7071232499767651215, 16294579238595022365, 33648306127698721183725, 527797716117331369424715
Offset: 1

Views

Author

Enoch Haga, Dec 25 2007

Keywords

Comments

This sequence is different from A070826 and A118750.
A clearer definition of the sequence: a(n) is the smallest odd composite number m such that m - 2 is prime and the set of the distinct prime factors of m equals the set of the first n odd primes. - Farideh Firoozbakht, Jun 30 2009

Examples

			The first odd prime is 3, 3*3-2 = 7 is prime so a(1) = 9.
The product of the first two odd primes is 15, and 15-2 is prime, so a(2) = 15.

Links

Crossrefs

Cf. A136349, A136350, A136351, A136352, A136354, A136355, A136356, A136357, A136358.
Cf. A070826, A118750.

Programs

  • Mathematica
    a[n_]:=(c=Product[Prime[k],{k,2,n+1}];For[m=1,!(!PrimeQ[c (2m-1)]&&PrimeQ[c(2m-1)-2]&&Length[FactorInteger[c(2m-1)]]==n),m++ ];c(2m-1));Table[a[n],{n,20}] (* Enoch Haga, Jul 02 2009 *)
  • PARI
    sm(n,x)=forprime(p=2,x, n/=p^valuation(n,p)); n==1
a(n)=my(m=factorback(primes(n+1)[2..n+1]),k,p=prime(n+1)); while(!isprime(k++*m-2) && sm(k,p),); k*m \\ Charles R Greathouse IV, Sep 14 2015

Formula

Compute N = product of the first n odd primes. If N-2 is prime, add N to the sequence. Otherwise test 3N, 5N, 7N, 9N, ... until kN - 2 is prime, subject to A006530(k) <= n+1.

Extensions

More terms, better title, and Mathematica program from Farideh Firoozbakht received Jun 30 2009. - Enoch Haga, Jul 02 2009
Further editing by Charles R Greathouse IV, Oct 05 2009
Showing 1-4 of 4 results.

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