

A280534


Number of partitions of n into two parts with the smaller part prime and the larger part squarefree.


2



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

1,8


COMMENTS

Number of distinct rectangles with squarefree length and prime width such that L + W = n, W <= L. For example, a(16) = 3; the rectangles are 2 X 14, 3 X 13 and 5 X 11.  Wesley Ivan Hurt, Nov 04 2017


LINKS

Table of n, a(n) for n=1..89.
Index entries for sequences related to partitions


FORMULA

a(n) = Sum_{i=1..floor(n/2)} A010051(i) * mu(ni)^2, where mu is the Möbius function (A008683).


MAPLE

with(numtheory): A280534:=n>add(mobius(ni)^2*(pi(i)pi(i1)), i=1..floor(n/2)): seq(A280534(n), n=1..100); # Wesley Ivan Hurt, Jan 04 2017


MATHEMATICA

Table[Sum[MoebiusMu[n  k]^2 * (PrimePi[k]  PrimePi[k  1]), {k, 1, Floor[n/2]}], {n, 1, 50}] (* G. C. Greubel, Jan 05 2017 *)


PROG

(PARI) for(n=1, 50, print1(sum(k=1, floor(n/2), isprime(k)*(moebius(nk))^2), ", ")) \\ G. C. Greubel, Jan 05 2017


CROSSREFS

Cf. A008683, A010051, A280535.
Sequence in context: A046799 A319506 A037809 * A129451 A097195 A274138
Adjacent sequences: A280531 A280532 A280533 * A280535 A280536 A280537


KEYWORD

nonn,easy


AUTHOR

Wesley Ivan Hurt, Jan 04 2017


STATUS

approved



