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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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4
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,6
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COMMENTS
| 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. [From T. D. Noe (noe(AT)sspectra.com), Oct 03 2008]
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REFERENCES
| D. M. Burton, Elementary Number Theory.
J. L. Brown, Proof of Scherk's Conjecture on the Representation of Primes, Amer. Math. Monthly 74 (1967), 31-33.
William Y. Lee, On the representation of integers, Math. Mag. 47 (1974), 150-152.
S. S. Pillai, "On some empirical theorem of Scherk", J. Indian Math. Soc. 17 (1927-28), pp. 164-171.
H. F. Scherk, "Bemerkungen über die Bildung der Primzahlen aus einander", Journal für die reine und angewandte Mathematik 10 (1883), pp. 201-208.
W. Sierpinski, Elementary Theory of Numbers, Warszawa, 1964.
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EXAMPLE
| 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
| 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;
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CROSSREFS
| 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.
A022894, A083309 [From T. D. Noe (noe(AT)sspectra.com), Oct 03 2008]
Sequence in context: A080013 A152949 A058660 * A076619 A007448 A155689
Adjacent sequences: A059868 A059869 A059870 * A059872 A059873 A059874
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KEYWORD
| nonn
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AUTHOR
| Antti Karttunen Feb 05 2001
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EXTENSIONS
| More terms from Naohiro Nomoto (n_nomoto(AT)yabumi.com), Sep 11 2001
More terms from Larry Reeves (larryr(AT)acm.org), Nov 20 2003
a(33)-a(39) from Donovan Johnson (donovan.johnson(AT)yahoo.com), Oct 01 2010
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