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A083309
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a(n) is the number of times that sums 3 +- 5 +- 7 +- 11 +- ... +- prime(2n+1) of the first 2n odd primes is zero. There are 2^(2n-1) choices for the sign patterns.
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27
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0, 0, 1, 2, 7, 19, 63, 197, 645, 2172, 7423, 25534, 89218, 317284, 1130526, 4033648, 14515742, 52625952, 191790090, 702333340, 2585539586, 9570549372, 35562602950, 131774529663, 491713178890, 1842214901398, 6909091641548
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OFFSET
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1,4
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COMMENTS
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The frequency of each possible sum is computed by the Mathematica program without explicitly computing the individual sums. Let S = 3 + 5 + 7 + ... + prime(2n+1). Because the primes do not grow very fast, it is easy to show that, for n > 2, all even numbers between -S+20 and S-20 occur at least once as a sum.
a(n) is the maximal number of subsets of {prime(2), prime(3), ..., prime(n+1)} that share the same sum. Cf. A025591, A083527.
See A238894 for a more general sequence that looks at all sums formed. - T. D. Noe, Mar 07 2014
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LINKS
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FORMULA
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EXAMPLE
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a(3) = 1 because there is only one sign pattern of the first six odd primes that yields zero: 3 + 5 + 7 - 11 + 13 - 17.
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MATHEMATICA
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d={1, 0, 0, 1}; nMax=32; zeroLst={}; Do[p=Prime[n+1]; d=PadLeft[d, Length[d]+p]+PadRight[d, Length[d]+p]; If[0==Mod[n, 2], AppendTo[zeroLst, d[[(Length[d]+1)/2]]]], {n, 2, nMax}]; zeroLst/2
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PROG
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(PARI) A083309(n, rhs=0, firstprime=2)={rhs-=prime(firstprime); my(p=vector(2*n-2+bittest(rhs, 0), i, prime(i+firstprime))); sum(i=1, 2^#p-1, sum(j=1, #p, (-1)^bittest(i, j-1)*p[j])==rhs)} \\ For illustrative purpose, too slow for n >> 10. - M. F. Hasler, Aug 08 2015
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CROSSREFS
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Cf. A022894 (use all primes in the sum), A022895 (r.h.s. = 1), A022896 (r.h.s. = 2), A022897 (interleaved 0 for odd number of terms), ..., A022903 (using primes >= 7), A022904, A022920; A261061 - A261063 and A261044 (r.h.s. = -1); A261057, A261059, A261060, A261045 (r.h.s. = -2).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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