

A047949


a(n) is the largest m such that nm and n+m are both primes, or 1 if no such m exists.


7



0, 0, 1, 2, 1, 4, 5, 4, 7, 8, 7, 10, 9, 8, 13, 14, 13, 12, 17, 16, 19, 20, 19, 22, 21, 20, 25, 24, 23, 28, 29, 28, 27, 32, 31, 34, 35, 34, 33, 38, 37, 40, 39, 38, 43, 42, 41, 30, 47, 46, 49, 50, 49, 52, 53, 52, 55, 54, 53, 48, 51, 50, 45, 62, 61, 64, 63, 62, 67, 68, 67, 66
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OFFSET

2,4


COMMENTS

A067076 is a subsequence of this sequence: when 2m+3 is prime a(m+3) = m. Moreover, it is the subsequence of records (maximal increasing subsequence): let m=a(n), with p=nm and q=p+2m both odd primes > 3; now 3+2(m+(p3)/2)=q and hence a(3+m+(p3)/2) >= m+(p3)/2 > m = a(n) but 3+m+(p3)/2 < n.  Jason Kimberley, Aug 30 2012 and Oct 10 2012
Goldbach's conjecture says a(n) >= 0 for all n.  Robert Israel, Apr 15 2015
a(n) is the Goldbach partition of 2n which results in the maximum spread divided by 2.  Robert G. Wilson v, Jun 18 2018


LINKS

T. D. Noe, Table of n, a(n) for n = 2..10000
OEIS (Plot 2), A067076 vs A098090 (nm=3).  Jason Kimberley, Oct 01 2012


FORMULA

a(n) = n  A020481(n).
a(n) = (A020482(n)  A020481(n))/2.  Gionata Neri, Apr 15 2015


EXAMPLE

4930=19 and 49+30=79 are primes, so a(49)=30.


MAPLE

a:= proc(n)
local k;
for k from n  1 to 0 by 2 do
if isprime(n+k) and isprime(nk) then return(k) fi
od:
1
end proc:
0, seq(a(n), n=3..1000); # Robert Israel, Apr 16 2015


MATHEMATICA

a[2] = a[3] = 0; a[n_] := (For[m = n  2, m >= 0, m, If[PrimeQ[n  m] && PrimeQ[n + m], Break[]]]; m); Table[a[n], {n, 2, 100}] (* JeanFrançois Alcover, Sep 04 2013 *)
lm[n_]:=Module[{m=n2}, While[!AllTrue[n+{m, m}, PrimeQ], m]; m]; Join[{0, 0}, Array[ lm, 70, 4]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 03 2014 *)
f[n_] := Block[{q = 2}, While[q <= n && !PrimeQ[2n q], q = NextPrime@ q]; n  q]; Array[f, 72, 2] (* Robert G. Wilson v, Jun 18 2018 *)


PROG

(PARI) a(n) = {if (n==2  n==3, return (0)); my(m = 1, lastm = 1, do = 1); while (do, if (isprime(nm) && isprime(n+m), lastm = m); m++; if (m == n  1, do = 0); ); return (lastm); } \\ Michel Marcus, Jun 09 2013
(PARI) a(n)=if(n<4, 0, forprime(p=3, n1, if(isprime(2*np), return(np))); 1) \\ Ralf Stephan, Dec 29 2013
(Haskell)
a047949 n = if null qs then 1 else head qs where
qs = [m  m < [n, n1 .. 0], a010051' (n+m) == 1, a010051' (nm) == 1]
 Reinhard Zumkeller, Nov 02 2015


CROSSREFS

Cf. A047160, A067076, A182138.
Cf. A020481.
Cf. A010051.
Sequence in context: A165050 A165042 A165046 * A222986 A222906 A323955
Adjacent sequences: A047946 A047947 A047948 * A047950 A047951 A047952


KEYWORD

easy,nice,nonn


AUTHOR

Lior Manor


EXTENSIONS

Corrected by Harvey P. Dale, Dec 21 2000


STATUS

approved



