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 1-5 of 5 results.

A230219 Number of ways to write 2*n + 1 = p + q + r with p <= q such that p, q, r are primes in A230217 and p + q + 9 is also prime.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 2, 5, 2, 2, 4, 3, 1, 4, 3, 1, 4, 1, 2, 5, 2, 3, 4, 3, 3, 8, 6, 3, 12, 6, 2, 13, 3, 3, 7, 6, 4, 5, 4, 4, 8, 7, 4, 12, 7, 3, 19, 6, 3, 16, 5, 4, 9, 5, 5, 7, 10, 4, 5, 8, 3, 14, 4, 3, 14, 2, 5, 12, 5, 2, 14, 9, 2, 10, 12, 4, 12, 7, 6, 12, 7, 9, 14, 8, 6, 12, 5, 4, 19, 8, 4, 23, 6, 3, 14
Offset: 1

Views

Author

Zhi-Wei Sun, Oct 11 2013

Keywords

Comments

Conjecture: a(n) > 0 for all n > 6.
This implies Goldbach's weak conjecture for odd numbers and also Goldbach's conjecture for even numbers.
The conjecture also implies that there are infinitely many primes in A230217.

Examples

			a(18) = 1 since 2*18 + 1 = 7 + 13 + 17, and 7, 13, 17 are terms of A230217, and 7 + 13 + 9 = 29 is a prime.

Links

Crossrefs

Cf. A002375, A068307, A230217, A230140, A230141.

Programs

  • Mathematica
    RQ[n_]:=PrimeQ[n+6]&&PrimeQ[3n+8]
SQ[n_]:=PrimeQ[n]&&RQ[n]
a[n_]:=Sum[If[RQ[Prime[i]]&&RQ[Prime[j]]&&PrimeQ[Prime[i]+Prime[j]+9]&&SQ[2n+1-Prime[i]-Prime[j]],1,0],{i,1,PrimePi[n-1]},{j,i,PrimePi[2n-2-Prime[i]]}]
Table[a[n],{n,1,100}]

A230223 Primes p such that 3*p-4, 3*p-10, and 3*p-14 are all prime.

Original entry on oeis.org

7, 11, 17, 19, 31, 37, 47, 59, 79, 107, 131, 151, 157, 229, 317, 367, 409, 431, 479, 499, 521, 541, 739, 787, 1031, 1181, 1307, 1381, 1487, 1601, 1697, 1747, 1951, 2551, 2749, 2767, 2917, 3251, 3391, 3449, 3581, 3931, 4217, 4349, 4447, 4567, 4639, 4721, 4909, 4967
Offset: 1

Views

Author

Zhi-Wei Sun, Oct 12 2013

Keywords

Comments

Conjecture: Any even number greater than 35 can be written as a sum of four terms of this sequence.
Primes in the sequence should be sparser than twin primes although this has not been proved.

Examples

			a(1) = 7 since 3*7-4 = 17, 3*7-10 = 11 and 3*7-14 = 7 are prime.

Links

Crossrefs

Cf. A023201, A046131, A230138, A230140, A230217, A230219, A230224.

Programs

  • Mathematica
    RQ[n_]:=n>5&&PrimeQ[3n-4]&&PrimeQ[3n-10]&&PrimeQ[3n-14]
m=0
Do[If[RQ[Prime[n]],m=m+1;Print[m," ",Prime[n]]],{n,1,1000}]
Select[Prime[Range[700]],AllTrue[3#-{4,10,14},PrimeQ]&] (* Harvey P. Dale, Sep 29 2023 *)
  • PARI
    is(p)=isprime(p) && isprime(3*p-4) && isprime(3*p-10) && isprime(3*p-14) \\ Charles R Greathouse IV, Oct 12 2013

A230230 Number of ways to write 2*n = p + q with p, q, 3*p - 10, 3*q + 10 all prime.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 3, 4, 3, 3, 5, 1, 5, 5, 3, 4, 5, 3, 2, 6, 4, 3, 6, 3, 3, 6, 3, 5, 6, 3, 6, 5, 4, 4, 9, 5, 4, 9, 5, 3, 9, 4, 4, 6, 4, 5, 6, 5, 5, 10, 4, 8, 10, 3, 7, 12, 3, 6, 11, 5, 7, 8, 3, 4, 6, 6, 4, 7, 2, 7, 9, 2, 10, 9, 3, 9, 8, 3, 5, 14, 8, 4, 12, 5, 5, 11, 5, 6, 8, 3, 8, 7, 4, 9, 11, 3
Offset: 1

Views

Author

Zhi-Wei Sun, Oct 12 2013

Keywords

Comments

Conjecture: a(n) > 0 for all n > 3.
This is stronger than Goldbach's conjecture for even numbers. If 2*n = p + q with p, q, 3*p - 10, 3*q + 10 all prime, then 6*n is the sum of the two primes 3*p - 10 and 3*q + 10.
Conjecture verified for 2*n up to 10^9. - Mauro Fiorentini, Jul 08 2023

Examples

			a(5) = 1 since 2*5 = 7 + 3 with 3*7 - 10 = 11 and 3*3 + 10 = 19 both prime.
a(16) = 1 since 2*16 = 13 + 19 with 3*13 - 10 = 29 and 3*19 + 10 = 67 both prime.

Links

Crossrefs

Cf. A002375, A187757, A199920, A227923, A230227, A230140, A230141, A230217, A230219, A230223, A230224.

Programs

  • Mathematica
    PQ[n_]:=n>3&&PrimeQ[3n-10]
SQ[n_]:=PrimeQ[n]&&PrimeQ[3n+10]
a[n_]:=Sum[If[PQ[Prime[i]]&&SQ[2n-Prime[i]],1,0],{i,1,PrimePi[2n-2]}]
Table[a[n],{n,1,100}]

A230227 Primes p with 3*p - 10 also prime.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 23, 31, 37, 41, 47, 53, 59, 61, 67, 79, 83, 89, 97, 101, 107, 109, 131, 137, 151, 157, 163, 167, 173, 191, 193, 199, 223, 229, 251, 257, 269, 277, 283, 307, 313, 317, 331, 347, 353, 367, 373, 397, 401, 409
Offset: 1

Views

Author

Zhi-Wei Sun, Oct 12 2013

Keywords

Comments

Conjecture: For any integer n > 4 not equal to 76, we have 2*n = p + q for some terms p and q from the sequence.
This is stronger than Goldbach's conjecture for even numbers.

Examples

			a(1) = 5 since 3*5 - 10 = 5 is prime.

Links

Crossrefs

Cf. A000040, A002375, A230138, A230140, A230141, A230217, A230219, A230223, A230224, A230230.

Programs

  • Mathematica
    PQ[p_]:=PQ[p]=p>3&&PrimeQ[3p-10]
m=0
Do[If[PQ[Prime[n]],m=m+1;Print[m," ",Prime[n]]],{n,1,80}]
Select[Prime[Range[100]],PrimeQ[3#-10]&] (* Harvey P. Dale, Jun 28 2015 *)

A230243 Number of primes p < n with 3*p + 8 and (p-1)*n + 1 both prime.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 1, 3, 1, 4, 2, 1, 4, 2, 2, 4, 2, 3, 2, 4, 3, 4, 4, 2, 2, 2, 1, 5, 3, 4, 3, 3, 2, 3, 4, 2, 2, 4, 2, 4, 4, 1, 5, 3, 2, 6, 4, 1, 5, 6, 3, 3, 5, 1, 5, 5, 2, 7, 5, 3, 4, 4, 3, 4, 6, 3, 4, 6, 4, 5, 6, 3, 7, 4, 2, 6, 1, 3, 5, 9, 3, 3, 7, 4, 3, 7, 1, 6, 5, 5, 5, 6, 3, 6, 7
Offset: 1

Views

Author

Zhi-Wei Sun, Oct 13 2013

Keywords

Comments

Conjecture: a(n) > 0 for all n > 4.
This implies A. Murthy's conjecture (cf. A034693) that for any integer n > 1, there is a positive integer k < n such that k*n + 1 is prime.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Sep 21 2023

Examples

			a(8) = 1 since 8 = 3 + 5 with 3, 3*3+8 = 17, (3-1)*8+1 = 17 all prime.
a(17) = 1 since 17 = 7 + 10, and 7, 3*7+8 = 29, (7-1)*17+1 = 103 are all prime.

Links

Crossrefs

Cf. A034693, A085053, A086685, A023210, A219864, A230217, A230219, A230241.

Programs

  • Mathematica
    a[n_]:=Sum[If[PrimeQ[3Prime[i]+8]&&PrimeQ[(Prime[i]-1)n+1],1,0],{i,1,PrimePi[n-1]}]
Table[a[n],{n,1,100}]
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.