A051034 Minimal number of primes needed to sum to n.
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
Offset: 2
Examples
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.
Links
- T. D. Noe, Table of n, a(n) for n = 2..10000
- Olivier Ramaré, On Šnirel'man's constant, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e série, 22:4 (1995), pp. 645-706.
- 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
Crossrefs
Programs
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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}] (* Jean-François Alcover, Apr 03 2012 *) -
PARI
issum(n,k)=if(k==1,isprime(n),k--;forprime(p=2,n,if(issum(n-p,k),return(1)));0) a(n)=my(k);while(!issum(n,k++),);k \\ Charles R Greathouse IV, Jun 01 2011
Formula
a(n) = 1 iff n is prime. a(2n) = 2 (for n > 1) if Goldbach's conjecture is true. a(2n+1) = 2 (for n >= 1) if 2n+1 is not prime, but 2n-1 is. a(2n+1) >= 3 (for n >= 1) if both 2n+1 and 2n-1 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
Assuming Goldbach's conjecture, a(n) <= 3. In particular, a(p)=1; a(2*n)=2 for n>1; a(p+2)=2 provided p+2 is not prime; otherwise a(n)=3. - Sean A. Irvine, Jul 29 2019
a(2n+1) <= 3 by Helfgott's proof of Goldbach's ternary conjecture, and hence a(n) <= 4 in general. - Charles R Greathouse IV, Oct 24 2022
Extensions
More terms from Naohiro Nomoto, Mar 16 2001