%I A059871
%S A059871 1,1,1,1,1,3,3,4,6,12,16,31,46,90,140,276,449,877,1443,2834,4725,9395,
%T A059871 16153,32037,55872,110288,190815,380488,672728,1342395,2434797,4808180
%N A059871 Number of solutions to the equation p_i = (1+mod(i,2))*p_{i-1} +- p_{i-2} 
               +- p_{i-3} +- ... +- 2 +- 1, where p_i is the i-th prime number (where 
               p_1 = 2 and the "zeroth prime" p_0 is defined to be 1).
%C A059871 In Burton's book it is said that it is "known" that each prime can be 
               represented as such sum. However, I do not know whether that means 
               it has been proved.
%C A059871 This is Scherk's theorem, which was conjectured by Scherk in 1833 and 
               proved by Pillai in 1928. [From T. D. Noe (noe(AT)sspectra.com), 
               Oct 03 2008]
%D A059871 D. M. Burton, Elementary Number Theory.
%D A059871 J. L. Brown, Proof of Scherk's Conjecture on the Representation of Primes, 
               Amer. Math. Monthly 74 (1967), 31-33. [From T. D. Noe (noe(AT)sspectra.com), 
               Oct 03 2008]
%D A059871 William Y. Lee, On the representation of integers, Math. Mag. 47 (1974), 
               150-152. [From T. D. Noe (noe(AT)sspectra.com), Oct 03 2008]
%D A059871 W. Sierpinski, Elementary Theory of Numbers, Warszawa, 1964. [From T. 
               D. Noe (noe(AT)sspectra.com), Oct 03 2008]
%e A059871 For the first five primes we have only one solution for each: 2 = 2*1, 
               3 = 1*2 + 1*1, 5 = 2*3 - 1*2 + 1*1, 7 = 1*5 + 1*3 - 1*2 + 1*1, 11 
               = 2*7 - 1*5 + 1*3 - 1*2 + 1*1 and for the next prime 13, we have 
               3 solutions: 13 = 11-7+5+3+2-1 = 11+7-5-3+2+1 = 11+7-5+3-2-1.
%p A059871 map(nops, primesums_primes_mult(16)); primesums_primes_mult := proc(upto_n) 
               local a,b,i,n,p,t; a := []; for n from 1 to upto_n do b := []; p 
               := ithprime(n); for i from (2^(n-1)) to ((2^n)-1) do t := bin_prime_sum(i); 
               if(t = p) then b := [op(b),i]; fi; od; a := [op(a),b]; print(a); 
               od; RETURN(a); end;
%Y A059871 See A059872 for the table of all solutions encoded as binary vectors 
               and A059873-A059875 for specific sequences. A059876 gives the function 
               bin_prime_sum.
%Y A059871 A022894, A083309 [From T. D. Noe (noe(AT)sspectra.com), Oct 03 2008]
%Y A059871 Sequence in context: A080013 A152949 A058660 this_sequence A076619 A007448 
               A155689
%Y A059871 Adjacent sequences: A059868 A059869 A059870 this_sequence A059872 A059873 
               A059874
%K A059871 nonn
%O A059871 1,6
%A A059871 Antti Karttunen Feb 05 2001
%E A059871 More terms from Naohiro Nomoto (n_nomoto(AT)yabumi.com), Sep 11 2001
%E A059871 More terms from Larry Reeves (larryr(AT)acm.org), Nov 20 2003