|
%I #33 Mar 15 2023 11:57:28
%S 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,
%T 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,
%U 6,9,4,10,5,9,6,10,5,9,5,9,6,9,5,12,6,8,6,11,8
%N Number of partitions of n into two parts with the smaller part prime and the larger part squarefree.
%C 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
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = Sum_{i=1..floor(n/2)} c(i) * mu(n-i)^2, where mu is the Möbius function (A008683) and c = A010051.
%p with(numtheory): A280534:=n->add(mobius(n-i)^2*(pi(i)-pi(i-1)), i=1..floor(n/2)): seq(A280534(n), n=1..100); # _Wesley Ivan Hurt_, Jan 04 2017
%t 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 *)
%t Table[Count[IntegerPartitions[n,{2}],_?(SquareFreeQ[#[[1]]]&&PrimeQ[ #[[2]]]&)],{n,90}] (* _Harvey P. Dale_, Oct 17 2021 *)
%o (PARI) for(n=1,50, print1(sum(k=1,floor(n/2), isprime(k)*(moebius(n-k))^2), ", ")) \\ _G. C. Greubel_, Jan 05 2017
%Y Cf. A008683, A010051, A280535.
%K nonn,easy
%O 1,8
%A _Wesley Ivan Hurt_, Jan 04 2017
|
