

A051034


Minimal number of primes needed to sum to n.


16



1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 1, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 2, 3, 2, 1, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 2, 3, 2, 3, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2
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OFFSET

2,3


LINKS

T. D. Noe, Table of n, a(n) for n = 2..10000
Yannick Saouter, Vinogradov's theorem is true up to 10^20
Terence Tao, Every odd number greater than 1 is the sum of at most five primes, arXiv:1201.6656 [math.NT], 2012 preprint, to appear in Mathematics of Computation.
Eric Weisstein's World of Mathematics, Prime Partition
Index entries for sequences related to Goldbach conjecture


FORMULA

a(n) = 1 iff n is prime. a(2n) = 2 (for n > 1) if Goldbachs's conjecture is true. a(2n+1) = 2 (for n >= 1) if 2n+1 is not prime, but 2n1 is. a(2n+1) >= 3 (for n >= 1) if both 2n+1 and 2n1 are not primes (for sufficiently large n, a(2n+1) = 3 by Vinogradov's theorem, 1937).  Franz Vrabec, Nov 30 2004
a(n) <= 3 for all n, assuming the Goldbach conjecture.  N. J. A. Sloane, Jan 20 2007
a(2n+1) <= 5, see Tao 2012.  Charles R Greathouse IV, Jul 09 2012


EXAMPLE

a(2) = 1 because 2 is already prime.
a(4) = 2 because 4 = 2+2 is a partition of 4 into 2 prime parts and there is no such partition with fewer terms.
a(27) = 3 because 27 = 3+5+19 is a partition of 27 into 3 prime parts and there is no such partition with fewer terms.


MATHEMATICA

(* Assuming Goldbach's conjecture *) a[p_?PrimeQ] = 1; a[n_] := If[ Reduce[ n == x + y, {x, y}, Primes] === False, 3, 2]; Table[a[n], {n, 2, 112}] (* JeanFrançois Alcover, Apr 03 2012 *)


PROG

(PARI) issum(n, k)=if(k==1, isprime(n), k; forprime(p=2, n, if(issum(np, k), return(1))); 0)
a(n)=my(k); while(!issum(n, k++), ); k \\ Charles R Greathouse IV, Jun 01 2011


CROSSREFS

Cf. A025583, A004526, A007944, A007962, A000607, A051036, A010051, A061358, A068307, A103765.
Different from A072491.
Sequence in context: A183025 A072410 A072491 * A024935 A082477 A036430
Adjacent sequences: A051031 A051032 A051033 * A051035 A051036 A051037


KEYWORD

nonn,nice


AUTHOR

Eric W. Weisstein


EXTENSIONS

More terms from Naohiro Nomoto, Mar 16 2001


STATUS

approved



