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A114374 Number of partitions of n into parts that are not squarefree. 6
1, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 3, 1, 0, 0, 5, 2, 2, 0, 7, 3, 2, 0, 11, 6, 4, 3, 15, 8, 6, 3, 22, 13, 11, 6, 34, 18, 15, 9, 46, 27, 24, 17, 64, 43, 33, 23, 89, 60, 51, 37, 124, 84, 78, 51, 166, 119, 109, 78, 226, 168, 152, 118, 300, 228, 215, 166, 404, 313, 300, 230, 546, 421, 409 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

a(A078135(n)) = 0; a(A078137(n)) > 0.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..10000

FORMULA

a(n) = A000041(n) - A073576(n) - A117395(n). - Reinhard Zumkeller, Mar 11 2006

G.f.: Product_{k>=1} (1 - mu(k)^2*x^k)/(1 - x^k), where mu(k) is the Moebius function (A008683). - Ilya Gutkovskiy, Dec 30 2016

EXAMPLE

a(12) = #{2*2*3, 2*2*2 + 2*2, 2*2 + 2*2 + 2*2} = 3;

a(13) = #{3*3 + 2*2} = 1.

MAPLE

with(numtheory):

b:= proc(n, i) option remember;

      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+

      `if`(i>n or issqrfree(i), 0, b(n-i, i))))

    end:

a:= n-> b(n$2):

seq(a(n), n=0..100);  # Alois P. Heinz, Jun 03 2015

MATHEMATICA

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n || SquareFreeQ[i], 0, b[n-i, i]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jun 30 2015, after Alois P. Heinz *)

PROG

(Haskell)

a114374 = p a013929_list where

   p _          0 = 1

   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m

-- Reinhard Zumkeller, Jun 01 2015

CROSSREFS

Cf. A013929, A073576.

Cf. A256012.

Sequence in context: A136263 A105593 A029371 * A111505 A109264 A322393

Adjacent sequences:  A114371 A114372 A114373 * A114375 A114376 A114377

KEYWORD

nonn

AUTHOR

Reinhard Zumkeller, Feb 09 2006

EXTENSIONS

Offset changed and a(0)=1 prepended by Reinhard Zumkeller, Jun 01 2015

STATUS

approved

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Last modified August 23 22:20 EDT 2019. Contains 326254 sequences. (Running on oeis4.)