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 1-3 of 3 results.

A047160 For n >= 2, a(n) = smallest number m >= 0 such that n-m and n+m are both primes, or -1 if no such m exists.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 3, 2, 3, 0, 1, 0, 3, 2, 3, 0, 1, 0, 3, 2, 9, 0, 5, 6, 3, 4, 9, 0, 1, 0, 9, 4, 3, 6, 5, 0, 9, 2, 3, 0, 1, 0, 3, 2, 15, 0, 5, 12, 3, 8, 9, 0, 7, 12, 3, 4, 15, 0, 1, 0, 9, 4, 3, 6, 5, 0, 15, 2, 3, 0, 1, 0, 15, 4, 3, 6, 5, 0, 9, 2, 15, 0, 5, 12, 3, 14, 9, 0, 7, 12, 9, 4, 15, 6, 7, 0, 9, 2, 3
Offset: 2

Views

Author

Lior Manor

Keywords

Comments

I have confirmed there are no -1 entries through integers to 4.29*10^9 using PARI. - Bill McEachen, Jul 07 2008
From Daniel Forgues, Jul 02 2009: (Start)
Goldbach's Conjecture: for all n >= 2, there are primes (distinct or not) p and q s.t. p+q = 2n. The primes p and q must be equidistant (distance m >= 0) from n: p = n-m and q = n+m, hence p+q = (n-m)+(n+m) = 2n.
Equivalent to Goldbach's Conjecture: for all n >= 2, there are primes p and q equidistant (distance >= 0) from n, where p and q are n when n is prime.
If this conjecture is true, then a(n) will never be set to -1.
Twin Primes Conjecture: there is an infinity of twin primes.
If this conjecture is true, then a(n) will be 1 infinitely often (for which each twin primes pair is (n-1, n+1)).
Since there is an infinity of primes, a(n) = 0 infinitely often (for which n is prime).
(End)
If n is composite, then n and a(n) are coprime, because otherwise n + a(n) would be composite. - Jason Kimberley, Sep 03 2011
From Jianglin Luo, Sep 22 2023: (Start)
a(n) < primepi(n)+sigma(n,0);
a(n) < primepi(primepi(n)+n);
a(n) < primepi(n), for n>344;
a(n) = o(primepi(n)), as n->+oo. (End)
If -1 < a(n) < n-3, then a(n) is divisible by 3 if and only if n is not divisible by 3, and odd if and only if n is even. - Robert Israel, Oct 05 2023

Examples

			16-3=13 and 16+3=19 are primes, so a(16)=3.

Links

Crossrefs

Cf. A001031, A002092, A002372, A002373, A002374, A002375, A014092, A025583, A035026, A047949, A071406, A082467, A102084, A103147, A112823, A155764, A155765, A177461, A078611, A010051, A045917, A325142.

Programs

  • Haskell
    a047160 n = if null ms then -1 else head ms
            where ms = [m | m <- [0 .. n - 1],
                            a010051' (n - m) == 1, a010051' (n + m) == 1]
-- Reinhard Zumkeller, Aug 10 2014
  • Magma
    A047160:=func;[A047160(n):n in[2..100]]; // Jason Kimberley, Sep 02 2011
  • Mathematica
    Table[k = 0; While[k < n && (! PrimeQ[n - k] || ! PrimeQ[n + k]), k++]; If[k == n, -1, k], {n, 2, 100}]
smm[n_]:=Module[{m=0},While[AnyTrue[n+{m,-m},CompositeQ],m++];m]; Array[smm,100,2] (* Harvey P. Dale, Nov 16 2024 *)
  • PARI
    a(n)=forprime(p=n,2*n, if(isprime(2*n-p), return(p-n))); -1 \\ Charles R Greathouse IV, Jun 23 2017
  • UBASIC
    10 N=2// 20 M=0// 30 if and{prmdiv(N-M)=N-M,prmdiv(N+M)=N+M} then print M;:goto 50// 40 inc M:goto 30// 50 inc N: if N>130 then stop// 60 goto 20

Formula

a(n) = n - A112823(n).
a(n) = A082467(n) * A005171(n), for n > 3. - Jason Kimberley, Jun 25 2012

Extensions

More terms from Patrick De Geest, May 15 1999
Deleted a comment. - T. D. Noe, Jan 22 2009
Comment corrected and definition edited by Daniel Forgues, Jul 08 2009

A307881 2*a(n) is the least number where k sets a new record such that 2*a(n)-k and 2*a(n)+k are prime and at least one of 2*a(n)-j and 2*a(n)+j is composite for all 0

Original entry on oeis.org

2, 4, 11, 23, 64, 68, 73, 119, 143, 172, 263, 452, 557, 868, 1238, 1579, 2864, 3533, 3637, 4252, 5171, 9263, 11282, 12388, 20036, 59119, 69332, 90131, 113783, 139283, 178612, 185714, 413788, 468059, 579932, 960707, 1879582, 2727031, 3266951, 3319868, 3591593
Offset: 1

Views

Author

Hugo Pfoertner, May 03 2019

Keywords

Comments

The corresponding records of k are given in A307882.

Links

Crossrefs

Cf. A065978, A293858, A307882, A325142.

Programs

  • PARI
    kmax=0; for(n=2,10^7,forstep(k=1,n,2,if(isprime(2*n-k)&&isprime(2*n+k),if(k>kmax,print1(n,", ");kmax=k);break(1))))

A307882 Records of the offset k in A307881, divided by 3.

Original entry on oeis.org

-1, 1, 3, 5, 7, 9, 11, 13, 15, 25, 29, 31, 39, 45, 61, 63, 75, 81, 85, 111, 123, 131, 151, 175, 207, 225, 241, 267, 301, 329, 335, 427, 435, 505, 539, 565, 611, 617, 665, 695, 739, 805, 843, 875, 1113, 1143, 1465, 1797, 1801, 1959, 2065, 2369, 2783
Offset: 1

Views

Author

Hugo Pfoertner, May 03 2019

Keywords

Comments

The divisibility of k by 3 of all terms after a(1) is conjectural. Offsets not divisible by 3 are given as -k instead of k/3.

Crossrefs

Cf. A065978, A293858, A307881, A325142.

Programs

  • PARI
    kmax=0; for(n=2,10^7,forstep(k=1,n,2,if(isprime(2*n-k)&&isprime(2*n+k),if(k>kmax,if(k%3!=0,print1(-k,", "),print1(k/3,", "));kmax=k);break(1))))
Showing 1-3 of 3 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.