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%I #64 Nov 23 2023 10:39:14
%S 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,
%T 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,
%U 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
%N 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.
%C I have confirmed there are no -1 entries through integers to 4.29*10^9 using PARI. - _Bill McEachen_, Jul 07 2008
%C From _Daniel Forgues_, Jul 02 2009: (Start)
%C 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.
%C 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.
%C If this conjecture is true, then a(n) will never be set to -1.
%C Twin Primes Conjecture: there is an infinity of twin primes.
%C If this conjecture is true, then a(n) will be 1 infinitely often (for which each twin primes pair is (n-1, n+1)).
%C Since there is an infinity of primes, a(n) = 0 infinitely often (for which n is prime).
%C (End)
%C If n is composite, then n and a(n) are coprime, because otherwise n + a(n) would be composite. - _Jason Kimberley_, Sep 03 2011
%C From _Jianglin Luo_, Sep 22 2023: (Start)
%C a(n) < primepi(n)+sigma(n,0);
%C a(n) < primepi(primepi(n)+n);
%C a(n) < primepi(n), for n>344;
%C a(n) = o(primepi(n)), as n->+oo. (End)
%C 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
%H T. D. Noe, <a href="/A047160/b047160.txt">Table of n, a(n) for n = 2..10000</a>
%H Jason Kimberley, <a href="/wiki/User:Jason_Kimberley/A047160">Symmetrical plot of A047160</a>
%F a(n) = n - A112823(n).
%F a(n) = A082467(n) * A005171(n), for n > 3. - _Jason Kimberley_, Jun 25 2012
%e 16-3=13 and 16+3=19 are primes, so a(16)=3.
%t Table[k = 0; While[k < n && (! PrimeQ[n - k] || ! PrimeQ[n + k]), k++]; If[k == n, -1, k], {n, 2, 100}]
%o (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
%o (Magma) A047160:=func<n|exists(r){m:m in[0..n-2]|IsPrime(n-m)and IsPrime(n+m)}select r else-1>;[A047160(n):n in[2..100]]; // _Jason Kimberley_, Sep 02 2011
%o (Haskell)
%o a047160 n = if null ms then -1 else head ms
%o where ms = [m | m <- [0 .. n - 1],
%o a010051' (n - m) == 1, a010051' (n + m) == 1]
%o -- _Reinhard Zumkeller_, Aug 10 2014
%o (PARI) a(n)=forprime(p=n,2*n, if(isprime(2*n-p), return(p-n))); -1 \\ _Charles R Greathouse IV_, Jun 23 2017
%Y Cf. A001031, A002092, A002372, A002373, A002374, A002375, A014092, A025583, A035026, A047949, A071406, A082467, A102084, A103147, A112823, A155764, A155765, A177461, A078611, A010051, A045917, A325142.
%K nonn,easy,nice
%O 2,7
%A _Lior Manor_
%E More terms from _Patrick De Geest_, May 15 1999
%E Deleted a comment. - _T. D. Noe_, Jan 22 2009
%E Comment corrected and definition edited by _Daniel Forgues_, Jul 08 2009
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