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A133364 Number of ways of writing n as a sum of a prime and a square-full number.

%I A133364 #18 Sep 13 2015 21:12:10
%S A133364 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,
%T A133364 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,
%U A133364 4,2,2,3,2,3,6,1,5,2,4,4,2,1,6,3,2,4,4,3
%N A133364 Number of ways of writing n as a sum of a prime and a square-full number.
%C A133364 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.
%H A133364 Nathaniel Johnston, <a href="/A133364/b133364.txt">Table of n, a(n) for n = 1..10000</a>
%F A133364 a(n) = Card{n = i + j where i is in A000040 and j is in A001694}.
%e A133364 a(3) = 1 because 3=2+1 where 2 is prime and 1 is square-full.
%e A133364 a(4) = 1 because 4=3+1 where 3 is prime and 1 is square-full.
%e A133364 a(5) = 0 because there is no positive solution to 5 = prime+(square-full).
%e A133364 a(6) = 2 because 6=5+1=2+4.
%e A133364 a(7) = 1 because 7=3+4.
%e A133364 a(8) = 1 because 8=7+1.
%e A133364 a(9) = 1 because 9=5+4.
%e A133364 a(10) = 1 because 10=2+8.
%e A133364 a(11) = 3 because 11=2+9=3+8=7+4.
%e A133364 a(12) = 2 because 12=3+9=11+1.
%e A133364 a(13) = 1 because 13=5+8.
%e A133364 a(14) = 2 because 14=5+9=13+1.
%e A133364 a(15) = 2 because 15=7+8=11+4.
%e A133364 a(16) = 1 because 16=7+9.
%e A133364 a(17) = 1 because 17=13+4.
%e A133364 a(18) = 2 because 18=2+16=17+1.
%e A133364 a(19) = 2 because 19=3+16=11+8.
%e A133364 a(20) = 2 because 20=19+1=11+9.
%p A133364 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
%t A133364 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 *)
%Y A133364 Cf. A000040, A000290, A001694, A002471, A005117, A098983.
%K A133364 easy,nonn
%O A133364 1,6
%A A133364 _Jonathan Vos Post_, Oct 26 2007
%E A133364 Corrected and extended by _Stefan Steinerberger_, Oct 29 2007 and by _R. J. Mathar_, Nov 09 2007