

A047160


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


16



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
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OFFSET

2,7


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 = nm and q = n+m, hence p+q = (nm)+(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 (n1, 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


LINKS

T. D. Noe, Table of n, a(n) for n=2..10000
J. S. Kimberley, Symmetrical plot of A047160


FORMULA

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


EXAMPLE

163=13 and 16+3=19 are primes, so a(16)=3.


MATHEMATICA

Table[k = 0; While[k < n && (! PrimeQ[n  k]  ! PrimeQ[n + k]), k++]; If[k == n, 1, k], {n, 2, 100}]


PROG

(UBASIC) 10 N=2// 20 M=0// 30 if and{prmdiv(NM)=NM, 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
(MAGMA) A047160:=func<nexists(r){m:m in[0..n2]IsPrime(nm)and IsPrime(n+m)}select r else1>; [A047160(n):n in[2..100]]; // Jason Kimberley, Sep 02 2011
(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


CROSSREFS

Cf. A001031, A002092, A002372, A002373, A002374, A002375, A014092, A025583, A035026, A047949, A071406, A082467, A102084, A103147, A112823, A155764, A155765, A177461.
Cf. A078611.  Stanislav Sykora, Mar 14 2014
Cf. A010051, A045917.
Sequence in context: A016458 A058513 A285736 * A093347 A230409 A244543
Adjacent sequences: A047157 A047158 A047159 * A047161 A047162 A047163


KEYWORD

nonn,easy,nice


AUTHOR

Lior Manor


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


STATUS

approved



