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.

Previous Showing 21-26 of 26 results.

A163152 Primes of the form PartitionsP[p], p are prime numbers.

Original entry on oeis.org

2, 3, 7, 101, 80630964769, 1394313503224447816939, 87674799670795146675673859587, 62607220478448273296879161314388228250413, 79074320470247928120049519839632230336234433216761019, 77355497906663686399579348109210219558359416885618588905259034616641337958059
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 21 2009

Keywords

Crossrefs

Cf. A098145, A163150, A163151, A049575.

Programs

  • Mathematica
    f[n_]:=PartitionsP[n]; lst={};Do[p=Prime[n];If[PrimeQ[f[p]],AppendTo[lst,f[p]]],{n,6!}];lst
Select[PartitionsP[Prime[Range[1000]]],PrimeQ] (* Harvey P. Dale, May 16 2020 *)
  • PARI
    forprime(p=2,1e4,k=numbpart(p);if(isprime(k),print1(k",")))

Extensions

Program by Charles R Greathouse IV, Oct 12 2009
More terms from Harvey P. Dale, May 16 2020

A234900 Primes p with P(p+1) also prime, where P(.) is the partition function (A000041).

Original entry on oeis.org

2, 3, 5, 131, 167, 211, 439, 2731, 3167, 3541, 4261, 7457, 8447, 18289, 22669, 23201, 23557, 35401, 44507, 76781, 88721, 108131, 126097, 127079, 136319, 141359, 144139, 159169, 164089, 177487, 202627, 261757, 271181, 282911, 291971, 307067, 320561, 389219, 481589, 482627, 602867, 624259, 662107, 682361, 818887, 907657, 914189, 964267, 1040191, 1061689
Offset: 1

Views

Author

Zhi-Wei Sun, Jan 01 2014

Keywords

Comments

It seems that this sequence contains infinitely many terms.
See also A234569 for a similar sequence.

Examples

			a(1) = 2 since P(2+1) = 3 is prime.
a(2) = 3 since P(3+1) = 5 is prime.
a(3) = 5 since P(5+1) = 11 is prime.

Links

Crossrefs

Cf. A000040, A000041, A049575, A233346, A234470, A234514, A234530, A234567, A234569, A234615, A234644.

Programs

  • Mathematica
    p[k_]:=p[k]=PrimeQ[PartitionsP[Prime[k]+1]]
n=0;Do[If[p[k],n=n+1;Print[n," ",Prime[k]]],{k,1,10000}]

A257662 Least prime q such that p(q*n) is prime, where p(.) is the partition function given by A000041.

Original entry on oeis.org

2, 2, 2, 47, 1481, 31, 11, 557, 277, 1847, 7, 3, 1861, 47, 1451, 557, 1429, 2, 18367, 2069, 13411, 463, 26731, 7, 50119, 61, 101, 877, 29, 11261, 2971, 421, 298589, 32633, 31, 55933, 5521, 7307, 22349, 11, 641, 13, 47881, 3, 2309, 51673, 94309, 186679, 136207, 1301
Offset: 1

Views

Author

Zhi-Wei Sun, Jul 12 2015

Keywords

Comments

Conjecture: a(n) exists for any n > 0.
This implies the conjecture that the sequence p(n) (n = 1,2,3,...) contains infinitely many primes.

Examples

			a(1) = 2 since p(2*1) = 2 is prime.
a(4) = 47 since 47 and p(47*4) = p(188) = 1398341745571 are both prime.

References

  • Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Links

Crossrefs

Cf. A000040, A000041, A046063, A049575, A259764.

Programs

  • Mathematica
    Do[k=0;Label[bb];k=k+1;If[PrimeQ[PartitionsP[Prime[k]*n]],Goto[aa],Goto[bb]]; Label[aa];Print[n, " ", Prime[k]];Continue,{n,1,50}]
  • PARI
    a(n)={my(r=1); while(!isprime(numbpart(prime(r)*n)), r++); return(prime(r));}
main(size)={return(vector(size,n,a(n)));} /* Anders Hellström, Jul 12 2015 */

A307547 a(n) is the smallest divisor of the partition number P(n) = A000041(n) not already in the sequence.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 15, 22, 6, 14, 4, 77, 101, 9, 8, 21, 27, 35, 10, 19, 12, 167, 251, 25, 89, 28, 43, 13, 55, 467, 311, 23, 49, 1231, 33, 17977, 281, 121, 45, 42, 193, 2417, 71, 31, 41, 73, 38, 7013, 275, 9283, 363, 53, 63, 17, 142, 47, 102359, 20, 44, 139
Offset: 1

Views

Author

Rémy Sigrist, Jul 27 2019

Keywords

Comments

Provided A046641(m) is defined for any number m > 0, this sequence is a permutation of the natural numbers.

Examples

			The first terms, alongside the divisors of P(n), are:
  n   a(n)  div(P(n))
  --  ----  --------------------
   1     1  (1)
   2     2  (1, 2)
   3     3  (1, 3)
   4     5  (1, 5)
   5     7  (1, 7)
   6    11  (1, 11)
   7    15  (1, 3, 5, 15)
   8    22  (1, 2, 11, 22)
   9     6  (1, 2, 3, 5, 6, 10, 15, 30)
  10    14  (1, 2, 3, 6, 7, 14, 21, 42)

Links

Crossrefs

Cf. A000041, A046641, A049575, A309200.

Programs

  • PARI
    See Links section.

Formula

a(A049575(k)) = A049575(k).

A163153 Primes of the form A000009(q)+q, where q are also prime numbers.

Original entry on oeis.org

3, 5, 23, 31, 73, 127, 797, 1301, 9851, 40099, 345953, 570181, 281138239, 48626519377, 91840127431010423, 130050574409983361, 6162297935619708167, 604490895403729930283, 3819342674540204978827, 20395819231612037821523
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 21 2009

Keywords

Comments

A subset of A121558.
Generated by positions q= 2, 3, 11, 13, 19, 23, 37, 41, 59, 73, 97..

Crossrefs

Cf. A098145, A163150, A163151, A049575, A163152.

Programs

  • Mathematica
    f[n_]:=PartitionsQ[n]+n; lst={};Do[p=Prime[n];If[PrimeQ[f[p]],AppendTo[lst, f[p]]],{n,6!}];lst
Select[Table[PartitionsQ[n]+n,{n,Prime[Range[300]]}],PrimeQ] (* Harvey P. Dale, Jun 02 2014 *)

Extensions

Edited by R. J. Mathar, Jul 25 2009

A368297 Prime plane partition numbers.

Original entry on oeis.org

3, 13, 859, 5668963, 12733429, 281846923, 10499640707, 776633557947931, 59206066030052023, 13621664240071959464038764694637, 27217095019687611064080107410267607999874139, 208912772327685894433117242327777497768893400876928857463950152067659
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 20 2023

Keywords

Comments

Prime values of A000219.

Links

Crossrefs

Cf. A000219, A049575, A051005, A285216 (indices).

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*numtheory[sigma][2](j), j=1..n)/n)
    end:
select(isprime, [seq(a(n), n=0..800)])[];  # Alois P. Heinz, Dec 20 2023
  • Mathematica
    nmax = 750; Select[CoefficientList[Series[Product[1/(1 - x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x], PrimeQ]

Formula

a(n) = A000219(A285216(n)).
Previous Showing 21-26 of 26 results.

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