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 A280535 Number of partitions of n into two parts with the smaller part squarefree and the larger part prime. 2
 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 (list; graph; refs; listen; history; text; internal format)
 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 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: A335875 A107279 A078461 * A221171 A333688 A319610 Adjacent sequences:  A280532 A280533 A280534 * A280536 A280537 A280538 KEYWORD nonn,easy AUTHOR Wesley Ivan Hurt, Jan 04 2017 STATUS approved

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Last modified August 15 12:56 EDT 2020. Contains 336502 sequences. (Running on oeis4.)