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 A262868 Number of squarefree numbers appearing among the larger parts of the partitions of n into two parts. 6
 0, 1, 1, 2, 1, 2, 2, 3, 3, 3, 3, 4, 3, 4, 4, 5, 5, 6, 6, 7, 6, 7, 7, 8, 8, 8, 8, 8, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 14, 14, 14, 15, 15, 15, 15, 16, 15, 16, 16, 17, 17, 18, 18, 19, 18, 19, 19, 19, 19, 20, 20, 21, 20, 21, 21, 22, 22 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Number of distinct rectangles with squarefree length and integer width such that L + W = n, W <= L. For example, a(14) = 4; the rectangles are 1 X 13, 3 X 11, 4 X 10 and 7 X 7. - Wesley Ivan Hurt, Nov 02 2017 a(10) = 3, a(100) = 30, a(10^3) = 302, a(10^4) = 3041, a(10^5) = 30393, a(10^6) = 303968, a(10^7) = 3039658, a(10^8) = 30396350, a(10^9) = 303963598, a(10^10) = 3039635373, a(10^11) = 30396355273, a(10^12) = 303963551068, a(10^13) = 3039635509338, a(10^14) = 30396355094469, a(10^15) = 303963550926043, a(10^16) = 3039635509271763, a(10^17) = 30396355092700721, and a(10^18) = 303963550927014110. The limit of a(n)/n is 3/Pi^2. - Charles R Greathouse IV, Nov 04 2017 LINKS Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 150 terms from G. C. Greubel) FORMULA a(n) = Sum_{i=1..floor(n/2)} mu(n-i)^2, where mu is the Möbius function A008683. a(n) = A262991(n) - A262869(n). a(n) ~ 3*n/Pi^2. - Charles R Greathouse IV, Nov 04 2017 EXAMPLE a(4)=2; there are two partitions of 4 into two parts: (3,1) and (2,2). Both of the larger parts are squarefree, thus a(4)=2. a(5)=1; there are two partitions of 5 into two parts: (4,1) and (3,2). Among the larger parts, only 3 is squarefree, thus a(5)=1. MAPLE with(numtheory): A262868:=n->add(mobius(n-i)^2, i=1..floor(n/2)): seq(A262868(n), n=1..100); MATHEMATICA Table[Sum[MoebiusMu[n - i]^2, {i, Floor[n/2]}], {n, 100}] Table[Count[IntegerPartitions[n, {2}][[All, 1]], _?SquareFreeQ], {n, 80}] (* Harvey P. Dale, Jan 03 2022 *) PROG (PARI) a(n) = sum(i=1, n\2, moebius(n-i)^2); \\ Michel Marcus, Oct 04 2015 (PARI) f(n)=my(s); forfactored(k=1, sqrtint(n), s+=n\k^2*moebius(k)); s a(n)=n--; f(n) - f(n\2) \\ Charles R Greathouse IV, Nov 04 2017 CROSSREFS Cf. A008683, A071068, A261985, A262351, A262869, A262870, A262871, A262991, A262992, A013928. Sequence in context: A333708 A132203 A158925 * A342097 A259357 A031265 Adjacent sequences:  A262865 A262866 A262867 * A262869 A262870 A262871 KEYWORD nonn,easy AUTHOR Wesley Ivan Hurt, Oct 03 2015 STATUS approved

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Last modified October 6 21:32 EDT 2022. Contains 357270 sequences. (Running on oeis4.)