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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 (list; graph; refs; listen; history; text; internal format)
 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 Nathaniel Johnston, Table of n, a(n) for n = 1..10000 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) ; 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 Cf. A000040, A000290, A001694, A002471, A005117, A098983. Sequence in context: A124010 A212171 A196228 * A063420 A254631 A029385 Adjacent sequences:  A133361 A133362 A133363 * A133365 A133366 A133367 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

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Last modified July 13 16:22 EDT 2020. Contains 335688 sequences. (Running on oeis4.)