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Search: id:A059871
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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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+0
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
(list; graph; listen)
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OFFSET
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1,6
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COMMENT
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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. [From T. D. Noe (noe(AT)sspectra.com), Oct 03 2008]
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REFERENCES
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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. [From T. D. Noe (noe(AT)sspectra.com), Oct 03 2008]
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]
W. Sierpinski, Elementary Theory of Numbers, Warszawa, 1964. [From T. D. Noe (noe(AT)sspectra.com), Oct 03 2008]
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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;
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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.
A022894, A083309 [From T. D. Noe (noe(AT)sspectra.com), Oct 03 2008]
Sequence in context: A080013 A152949 A058660 this_sequence A076619 A007448 A155689
Adjacent sequences: A059868 A059869 A059870 this_sequence A059872 A059873 A059874
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KEYWORD
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nonn
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AUTHOR
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Antti Karttunen Feb 05 2001
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EXTENSIONS
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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
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