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A098983 Number of ways of writing n as a sum of a prime and a squarefree number. 8
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; text; internal format)
OFFSET

0,5

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."

LINKS

T. D. Noe, Table of n, a(n) for n = 0..10000

Adrian Dudek, On the Sum of a Prime and a Square-free Number, arXiv:1410.7459 [math.NT], 2014.

T. Estermann, On the representations of a number as the sum of a prime and a quadratfrei number, J. London Math. Soc., S1-6(3):219, 1931.

A. Page, On the Number of Primes in an Arithmetic Progression, Proc London Math Soc (1935) s2-39 (1): 116-141.

A. Walfisz, Zur additiven Zahlentheorie II, Math. Z. 40 (1936), 592-607.

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. [Reinhard Zumkeller, Nov 04 2009]

EXAMPLE

a(8) = 4: 8=2+6=3+5=5+3=7+1.

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] (* Jean-Fran├žois Alcover, Jul 20 2011 *)

PROG

(Haskell)

a098983 n = sum $ map (a008966 . (n -)) $ takeWhile (< n) a000040_list

-- Reinhard Zumkeller, Sep 14 2011

(PARI) a(n)=my(s); forprime(p=2, n, s+=issquarefree(n-p)); s \\ Charles R Greathouse IV, Jun 20 2013

CROSSREFS

Sequence in context: A153437 A023153 A023159 * A097576 A029250 A110884

Adjacent sequences:  A098980 A098981 A098982 * A098984 A098985 A098986

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane, Oct 24 2004

STATUS

approved

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Last modified February 21 00:32 EST 2018. Contains 299388 sequences. (Running on oeis4.)