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

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

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)[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

Cf. A000040, A000290, A001694, A002471, A005117, A098983.

Sequence in context: A337255 A337375 A196228 * A063420 A347917 A254631

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