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.

A038771 a(n) is the smallest composite number c such that A002110(n) + c is prime.

Original entry on oeis.org

4, 9, 25, 49, 121, 221, 289, 529, 667, 899, 1147, 1591, 2021, 1849, 2773, 3551, 4087, 4819, 4757, 5041, 7519, 7663, 8549, 9991, 10379, 13231, 11227, 14659, 11881, 21877, 25283, 18209, 22331, 20989, 22499, 25591, 27221, 29503, 31313, 34547
Offset: 0

Views

Author

Labos Elemer, May 04 2000

Keywords

Comments

The lower "envelope" of the sequence is prime(n+1)^2. See also Fortune-conjecture (A005235).
For some n, c=prime(n+1)^2; for others, it is larger, even not necessarily divisible by prime(n+1). E.g., at n=11, prime(11)=31 and a(11) = 1591 = 37*43 = prime(12)*prime(14), while for n=59, a(59) = 97969 = 313^2 = prime(65)^2, etc. Adding these to the suitable primorial numbers, primes are obtained.
Conjecture: lim inf_{n->oo} a(n)/prime(n+1)^2 = 1 < lim sup_{n->oo} a(n)/prime(n+1)^2 = 2. - Charles R Greathouse IV and Thomas Ordowski, Apr 24 2015
Conjecture: all the terms in this sequence have exactly two prime factors. This conjecture is true for the first 133 terms. - Dmitry Kamenetsky, Jan 06 2019

Links

Crossrefs

Cf. A002110, A054757, A054758, A005235.

Programs

  • PARI
    a(n) = {my(q = prod(i=1, n, prime(i))); forcomposite(c = 1,, if (isprime(q+c), return(c);););} \\ Michel Marcus, May 24 2015

Extensions

Name edited by Tom Edgar, Jun 08 2015
a(0) prepended by Dmitry Kamenetsky, Jan 06 2019

A038773 a(n) is the smallest prime of the form Q + c, where Q is the n-th primorial and c is a composite >= prime(n+1)^2.

Original entry on oeis.org

11, 31, 79, 331, 2531, 30319, 511039, 9700357, 223093769, 6469694377, 200560491721, 7420738136831, 304250263529059, 13082761331672803, 614889782588494961, 32589158477190048817, 1922760350154212643889, 117288381359406970988027, 7858321551080267055884131, 557940830126698960967422909
Offset: 1

Views

Author

Labos Elemer, May 04 2000

Keywords

Comments

Between 2310 and 2531 there are 26 primes (2311, ..., 2521), all of which are of the form (primorial + prime). (2311 = 2 + 2309 (prime) = 2*3*5 + 2281 (prime); each of the other 25 primes is of the form 2*3*5*7*11 + prime.)
Observe that a(2) = 31 = 2*3 + 5^2 = 2*3*5 + 1, so it has two "primorial forms".

Examples

			At n=5, the 5th primorial is A002110(5)=2310 and 2310 + 13*17 = 2310 + 221 = 2531 is the prime that meets the criteria of the definition.

Links

Crossrefs

Cf. A002110, A054757, A054758, A005235.

Programs

  • Mathematica
    Array[Block[{Q = Product[Prime@ i, {i, #}], c = Prime[# + 1]^2}, While[Nand[PrimeQ[Q + c], CompositeQ@ c], c++]; Q + c] &, 17] (* Michael De Vlieger, May 22 2018 *)
  • PARI
    a(n) = {my(pr = prod(k=1, n, prime(k)), c = prime(n+1)^2); while (isprime(c) || !isprime(pr + c), c++); pr + c;} \\ Michel Marcus, May 26 2018

Extensions

Edited by Jon E. Schoenfield, May 22 2018
More terms from Michael De Vlieger, May 22 2018

A038767 Numbers k for which k-th primorial + square of (k+1)-th prime is also a prime.

Original entry on oeis.org

1, 2, 3, 4, 6, 13, 19, 28, 41, 66, 85, 371, 437, 726, 924, 1063, 3401, 6714
Offset: 1

Views

Author

Labos Elemer, May 04 2000

Keywords

Comments

r = prime(n+1)^2 is the smallest possible composite number that, if added to the n-th primorial, might give a prime.

Examples

			a(5)=6, 6th primorial is 30030, square of 7th prime is 289, sum gives 30319, a prime.

Crossrefs

Cf. A002110, A054758, A106155, A245694.

Programs

  • Mathematica
    Block[{a = {}, p = 1, q = 1}, Do[q = NextPrime[q]; If[PrimeQ[p + q^2], AppendTo[a, i]]; p *= q, {i, 1200}]; Rest[a] - 1] (* Michael De Vlieger, Jan 03 2021 *)
  • PARI
    isok(n) = ispseudoprime(prime(n+1)^2 + prod(j=1, n, prime(j))); \\ Michel Marcus, Aug 26 2019

Formula

a(n) = A002110(n) + prime(n+1)^2 is prime; n so that A054758(n)=1.

Extensions

a(12)-a(14) from Michel Marcus, Aug 26 2019
a(15)-a(16) (due to Jon E. Schoenfield at A245694) from Bill McEachen, Jan 03 2021
a(17) from Michael S. Branicky, Jun 09 2023
a(18) from Michael S. Branicky, Aug 09 2024

A038768 Primes of the form A002110(k) + prime(k+1)^2.

Original entry on oeis.org

11, 31, 79, 331, 30319, 304250263529059, 7858321551080267055884131, 2566376117594999414479597815340071648406351, 29819592777931214269172453467810429868925511217482600306406141434190851
Offset: 1

Views

Author

Labos Elemer, May 04 2000

Keywords

Examples

			For the 6th term, n=13 and 304250263529059 = 2*3*5*7*11*13*17*19*23*29*31*37*41 + 43^2.

Crossrefs

Cf. A002110, A054758, A054757.
Showing 1-4 of 4 results.

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