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Search: id:A083309
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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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+0
5
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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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COMMENT
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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.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..100
T. D. Noe, Extremal Sums of Sequences
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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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CROSSREFS
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Cf. A015818, A063865.
Cf. A022894 (sums of all primes)
Adjacent sequences: A083306 A083307 A083308 this_sequence A083310 A083311 A083312
Sequence in context: A030224 A114624 A091024 this_sequence A080873 A126162 A054423
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KEYWORD
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easy,nonn
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AUTHOR
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T. D. Noe (noe(AT)sspectra.com), Apr 29 2003
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