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A054845
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Number of ways of representing n as the sum of one or more consecutive primes.
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12
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0, 0, 1, 1, 0, 2, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 2, 1, 1, 0, 0, 0, 2, 1, 0, 1, 0, 1, 1, 1, 2, 0, 0, 0, 0, 2, 1, 0, 1, 0, 3, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 2, 0, 0, 1, 0, 1, 2, 2, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 2, 2, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 3, 1, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, 1, 0, 2, 1, 0, 2, 2
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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COMMENTS
| Moser shows that the average order of a(n) is log 2, that is, sum(i=1..n, a(i)) ~ n log 2. This shows that a(n) = 0 infinitely often (and with positive density); Moser asks if a(n) = 1 infinitely often, if a(n) = k is solvable for all k, whether these have positive density, and whether the sequence is bounded. [Charles R Greathouse IV, Mar 21, 2011]
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REFERENCES
| R. K. Guy, Unsolved Problems In Number Theory, C2.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..10000
Leo Moser, Notes on number theory. III. On the sum of consecutive primes, Canad. Math. Bull. 6 (1963), pp. 159-161.
C. Rivera, Prime Puzzles
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EXAMPLE
| a(5)=2 because of 2+3 and 5. a(17)=2 because of 2+3+5+7 and 17.
41 = 41 = 11+13+17 = 2+3+5+7+11+13, so a(41)=3.
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MATHEMATICA
| f[n_] := Block[{p = Prime@ Range@ PrimePi@ n}, len = Length@ p; Count[(Flatten[#, 1] &)[Table[ p[[i ;; j]], {i, len}, {j, i, len}]], q_ /; Total@ q == n]]; f[0] = 0; Array[f, 102, 0](* Jean-François Alcover, Feb 16 2011*) (* fixed by RGWv *)
nn=100; p=Prime[Range[PrimePi[nn]]]; t=Table[0, {nn}]; Do[s=0; j=i; While[s=s+p[[j]]; s<=nn, t[[s]]++; j++], {i, Length[p]}]; Join[{0}, t]
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PROG
| (PARI){/* program gives nn+1 values of a(n) for n=0..nn */
nn=2000; t=vector(nn+1); forprime(x=2, nn, s=x;
forprime(y=x+1, nn, if(s<=nn, t[s+1]++, break()); s=s+y)); t} [Zak Seidov, Feb 17 2011]
(MAGMA) S:=[0]; for n in [1..104] do count:=0; for q in PrimesUpTo(n) do p:=q; s:=p; while s lt n do p:=NextPrime(p); s+:=p; end while; if s eq n then count+:=1; end if; end for; Append(~S, count); end for; S; // Klaus Brockhaus, Feb 17 2011
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CROSSREFS
| Cf. A000586, A054859.
Sequence in context: A141571 A164067 A113706 * A117163 A096863 A117210
Adjacent sequences: A054842 A054843 A054844 * A054846 A054847 A054848
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KEYWORD
| nice,nonn
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AUTHOR
| Jud McCranie (JudMcCranie(AT)ugaalum.uga.edu), May 25 2000
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Oct 27 2008 at the suggestion of Jake M. Foster.
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