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A133364
Number of ways of writing n as a sum of a prime and a square-full number.
3
0, 0, 1, 1, 0, 2, 1, 1, 1, 1, 3, 2, 1, 2, 2, 1, 1, 2, 2, 2, 3, 1, 2, 1, 1, 1, 4, 2, 2, 3, 1, 4, 2, 2, 3, 1, 2, 5, 4, 2, 2, 2, 2, 3, 4, 2, 3, 2, 3, 2, 4, 2, 2, 3, 3, 4, 2, 1, 2, 2, 2, 4, 3, 1, 2, 3, 3, 5, 4, 2, 2, 3, 2, 3, 6, 1, 5, 2, 4, 4, 2, 1, 6, 3, 2, 4, 4, 3
OFFSET
1,6
COMMENTS
This is to square-full numbers A001694 as A098983 is to squarefree numbers A005117 and as A002471 is to squares A000290. Asymptotics of this should relate to A098983.
LINKS
FORMULA
a(n) = Card{n = i + j where i is in A000040 and j is in A001694}.
EXAMPLE
a(3) = 1 because 3=2+1 where 2 is prime and 1 is square-full.
a(4) = 1 because 4=3+1 where 3 is prime and 1 is square-full.
a(5) = 0 because there is no positive solution to 5 = prime+(square-full).
a(6) = 2 because 6=5+1=2+4.
a(7) = 1 because 7=3+4.
a(8) = 1 because 8=7+1.
a(9) = 1 because 9=5+4.
a(10) = 1 because 10=2+8.
a(11) = 3 because 11=2+9=3+8=7+4.
a(12) = 2 because 12=3+9=11+1.
a(13) = 1 because 13=5+8.
a(14) = 2 because 14=5+9=13+1.
a(15) = 2 because 15=7+8=11+4.
a(16) = 1 because 16=7+9.
a(17) = 1 because 17=13+4.
a(18) = 2 because 18=2+16=17+1.
a(19) = 2 because 19=3+16=11+8.
a(20) = 2 because 20=19+1=11+9.
MAPLE
isA001694 := proc(n) local digs, i ; digs := ifactors(n)[2] ; for i in digs do if op(2, i) = 1 then RETURN(false) ; fi ; od: RETURN(true) ; end: A133364 := proc(n) local a, p ; a := 0 ; p := 2 ; while p < n do if isA001694(n-p) then a := a+1 ; fi ; p := nextprime(p) ; od: RETURN(a) ; end: seq(A133364(n), n=3..90) ; # R. J. Mathar, Nov 09 2007
MATHEMATICA
a = {}; For[n = 3, n < 100, n++, c = 0; For[j = 1, Prime[j] < n, j++, d = 1; b = FactorInteger[n - Prime[j]]; For[m = 1, m < Length[b] + 1, m++, If[b[[m, 2]] < 2, d = 0]]; If[d == 1, c++ ]]; AppendTo[a, c]]; a (* Stefan Steinerberger, Oct 29 2007 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Oct 26 2007
EXTENSIONS
Corrected and extended by Stefan Steinerberger, Oct 29 2007 and by R. J. Mathar, Nov 09 2007
STATUS
approved