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A262992
Sum of the squarefree numbers among the partition parts of n into two parts.
6
0, 2, 3, 8, 6, 14, 17, 24, 24, 29, 34, 51, 45, 65, 72, 87, 87, 104, 104, 133, 123, 155, 166, 189, 189, 202, 215, 229, 215, 259, 274, 305, 305, 355, 372, 407, 407, 463, 482, 521, 521, 583, 604, 669, 647, 670, 693, 740, 740, 740, 740, 817, 791, 844, 844, 899
OFFSET
1,2
FORMULA
a(n) = Sum_{i=1..floor(n/2)} i*mu(i)^2 + (n-i)*mu(n-i)^2, where mu is the Möebius function (A008683).
a(n) = A262870(n) + A262871(n).
EXAMPLE
a(3)=3; there is one partition of 3 into two parts: (2,1). The sum of the squarefree parts of this partition is 2+1=3, so a(3)=3.
a(5)=6; there are two partitions of 5 into two parts: (4,1) and (3,2). The sum of the squarefree parts of these partitions is 3+2+1=6, so a(5)=6.
MAPLE
with(numtheory): A262992:=n->add(i*mobius(i)^2 + (n-i)*mobius(n-i)^2, i=1..floor(n/2)): seq(A262992(n), n=1..100);
MATHEMATICA
Table[Sum[i*MoebiusMu[i]^2 + (n - i)*MoebiusMu[n - i]^2, {i, Floor[n/2]}], {n, 100}]
PROG
(PARI) vector(100, n, sum(k=1, n\2, k*moebius(k)^2 + (n-k)*moebius(n-k)^2)) \\ Altug Alkan, Oct 07 2015
(PARI) a(n)=my(s, k2, m=n-1); forsquarefree(k=1, sqrtint(m), k2=k[1]^2; s+= k2*binomial(m\k2+1, 2)*moebius(k)); s + (n%4==2 && issquarefree(n/2))*n/2 \\ Charles R Greathouse IV, Jan 13 2018
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Oct 06 2015
STATUS
approved