A280535 Number of partitions of n into two parts with the smaller part squarefree and the larger part prime.

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

Offset

1,4

Comments

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

Links

Table of n, a(n) for n=1..90.

Index entries for sequences related to partitions

Formula

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).

Maple

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);

Mathematica

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

Prog

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

Crossrefs

Cf. A008683, A010051, A280534.

Sequence in context: A107279 A078461 A341945 * A221171 A333688 A319610

Adjacent sequences: A280532 A280533 A280534 * A280536 A280537 A280538

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Jan 04 2017

Status

approved