A280535 - OEIS

%I #33 May 02 2020 00:38:37

%S 0,0,1,2,1,2,1,2,1,2,0,2,2,3,1,2,1,3,2,3,2,3,2,4,2,3,1,2,2,4,2,4,3,5,

%T 1,4,2,4,2,3,1,5,3,6,3,5,2,6,2,5,3,5,3,6,3,4,3,6,2,7,3,6,4,6,2,6,4,5,

%U 4,6,3,8,5,9,4,8,4,8,3,8,6,9,3,9,6,8,3,7,5,10

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

%C Number of distinct rectangles with prime length and squarefree width such that L + W = n, W <= L. For example, a(14) = 3; the rectangles are 1 X 13, 3 X 11 and 7 X 7. - _Wesley Ivan Hurt_, Nov 18 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(n-i) * mu(i)^2, where c is the prime characteristic (A010051) and mu is the Möbius function (A008683).

%p with(numtheory): A280535:=n->add(mobius(i)^2*(pi(n-i)-pi(n-i-1)), i=1..floor(n/2)): seq(A280535(n), n=1..100);

%t Table[Sum[MoebiusMu[k]^2 * (PrimePi[n - k] - PrimePi[n - k - 1]), {k, 1, Floor[n/2]}], {n, 1, 50}] (* _G. C. Greubel_, Jan 05 2017 *)

%t Table[Count[IntegerPartitions[n,{2}],_?(PrimeQ[#[[1]]] && SquareFreeQ[ #[[2]]]&)],{n,90}] (* _Harvey P. Dale_, Feb 25 2018 *)

%o (PARI) for(n=1, 50, print1(sum(k=1, floor(n/2), isprime(n-k)*(moebius(k))^2), ", ")) \\ _G. C. Greubel_, Jan 05 2017

%Y Cf. A008683, A010051, A280534.

%K nonn,easy

%O 1,4

%A _Wesley Ivan Hurt_, Jan 04 2017