A294234 Number of partitions of n into two parts such that the smaller part is nonsquarefree.

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13

Offset

0,17

Links

Table of n, a(n) for n=0..78.

Index entries for sequences related to partitions

Formula

a(n) = Sum_{i=1..floor(n/2)} (1 - mu(i)^2), where mu is the Möbius function (A008683).

a(2*n) = a(2*n + 1) for n >= 0. - David A. Corneth, Oct 25 2017

Example

The nonsquarefree numbers up to 10 are 4, 8 and 9. So a(n) = 0 for n = 0 to 2*4 - 1 = 7, a(n) = 1 for n = 2*4 to 2*8 - 1, a(n) = 2 for n = 2*8 = 16 to 2*9 - 1 = 17. We haven't filled anything in yet for n = 18 to 2 * 10 = so a(n) = 3 for n = 18 to 20. We haven't checked for nonsquarefree numbers up for n > 10 so stop here. - David A. Corneth, Oct 25 2017

Mathematica

Table[Sum[(1 - MoebiusMu[k]^2), {k, Floor[n/2]}], {n, 200}]
Table[Count[IntegerPartitions[n, {2}], _?(!SquareFreeQ[#[[2]]]&)], {n, 0, 80}] (* Harvey P. Dale, Sep 18 2021 *)

Prog

(PARI) first(n) = {my(res = vector(n), nsqrfr = List(), t = 0); for(i = 2, sqrtint(n\2), for(k = 1, (n\2) \ i^2, listput(nsqrfr, k * i^2))); listsort(nsqrfr, 1); for(i = 1, #nsqrfr, for(j = t, nsqrfr[i] - 1, for(k = 1, 2, res[2*j + k] = i-1)); t = nsqrfr[i]); for(i=2*t+1, n, res[i] = res[2*t] + 1); res} \\ David A. Corneth, Oct 25 2017

Crossrefs

Cf. A008683, A008966, A013929, A294235.

Sequence in context: A157791 A236857 A156874 * A078767 A331137 A093125

Adjacent sequences: A294231 A294232 A294233 * A294235 A294236 A294237

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Oct 25 2017

Status

approved