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A059871
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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).
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5
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1, 1, 1, 1, 1, 3, 3, 4, 6, 12, 16, 31, 46, 90, 140, 276, 449, 877, 1443, 2834, 4725, 9395, 16153, 32037, 55872, 110288, 190815, 380488, 672728, 1342395, 2434797, 4808180, 8579625, 17070112, 30858078, 61271317, 110926277, 220979544, 402354848
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OFFSET
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1,6
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COMMENTS
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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.
This is Scherk's theorem, which was conjectured by Scherk in 1833 and proved by Pillai in 1928. [T. D. Noe, Oct 03 2008]
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REFERENCES
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D. M. Burton, Elementary Number Theory.
S. S. Pillai, "On some empirical theorem of Scherk", J. Indian Math. Soc. 17 (1927-28), pp. 164-171.
W. Sierpiński, Elementary Theory of Numbers, Warszawa, 1964.
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LINKS
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EXAMPLE
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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.
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MAPLE
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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;
# second Maple program
p:= n-> `if`(n<0, 0, `if`(n=0, 1, ithprime(n))):
sp:= proc(n) sp(n):= `if`(n<0, 0, p(n)+sp(n-1)) end:
b := proc(n, i) option remember; `if`(n>sp(i), 0, `if`(i<0, 1,
b(n+p(i), i-1)+ b(abs(n-p(i)), i-1)))
end:
a:= n-> b(p(n) -(1+irem(n, 2))*p(n-1), n-2):
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MATHEMATICA
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nmax = 40; d = {1}; a1 = {}; pp = 1;
Do[
p = Prime[n];
i = Ceiling[Length[d]/2] + Abs[p - (1 + Mod[n, 2])*pp];
AppendTo[a1, If[i > Length[d], 0, d[[i]]]];
d = PadLeft[d, Length[d] + 2 pp] + PadRight[d, Length[d] + 2 pp];
pp = p;
, {n, nmax}];
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CROSSREFS
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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.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Nov 20 2003
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STATUS
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approved
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