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A098983
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Number of ways of writing n as a sum of a prime and a squarefree number.
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2
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0, 0, 0, 1, 2, 2, 2, 2, 4, 3, 3, 1, 4, 4, 4, 3, 5, 4, 6, 4, 6, 4, 6, 3, 9, 5, 7, 3, 7, 4, 7, 4, 8, 7, 9, 4, 10, 6, 8, 6, 10, 6, 11, 7, 12, 8, 11, 5, 13, 8, 11, 6, 11, 8, 13, 6, 10, 7, 13, 6, 16, 7, 13, 8, 16, 7, 14, 7, 13, 10, 15, 7, 18, 10, 17, 10, 18, 9, 17, 8, 17, 12, 17, 8, 21, 12, 15, 9, 18, 13
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| From a posting by Hugh Montgomery to the Number Theory mailing list, Oct 05 2004: "Estermann, JLMS (1931), established an asymptotic formula for a(n). Page, PLMS (1935), gave a quantitative version of this, with an error term roughly (log n)^5 smaller than the main term. Walfisz, Zur additiven Zahlentheorie II, Math. Z. 40 (1936), 592-607, established what we know today as the "Siegel-Walfisz theorem" in a series of lemmas and used this new tool to give the formula for a(n) with an error term that is smaller by a factor (log n)^c for any c."
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LINKS
| T. D. Noe, Table of n, a(n) for n = 0..10000
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FORMULA
| G.f.: (x^2+x^3+x^5+x^7+x^11+x^13+x^17+x^19+...)(x+x^2+x^3+x^5+x^6+x^7+x^10+x^11+x^13+x^14+x^15+x^17+x^19+...).
a(n+1) = SUM(A008966(k)*A010051(n-k+1): 1<=k<=n) for n>0. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 04 2009]
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EXAMPLE
| a(8) = 4: 8=2+6=3+5=5+3=7+1.
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MATHEMATICA
| m = 90; sf = Total[ x^Select[Range[m], SquareFreeQ] ]; pp = Sum[x^Prime[n], {n, 1, PrimePi @ Exponent[sf[[-1]], x]}]; CoefficientList[Series[pp * sf, {x, 0, m-1}], x] (* From Jean-François Alcover, Jul 20 2011 *)
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PROG
| (Haskell)
a098983 n = sum $ map (a008966 . (n -)) $ takeWhile (< n) a000040_list
-- Reinhard Zumkeller, Sep 14 2011
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CROSSREFS
| Sequence in context: A153437 A023153 A023159 * A097576 A029250 A110884
Adjacent sequences: A098980 A098981 A098982 * A098984 A098985 A098986
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KEYWORD
| nonn,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Oct 24 2004
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