A279910 - OEIS

%I #12 Dec 24 2016 09:20:01

%S 0,0,3,3,8,11,18,18,21,26,37,40,53,60,68,68,85,88,107,112,122,133,156,

%T 159,164,177,180,187,216,224,255,255,269,286,298,301,338,357,373,378,

%U 419,429,472,483,491,514,561,564,571,576,596,609,662,665,681,688,710,739,798,806,867,898,908,908,926,940,1007,1024,1050,1062

%N a(n) = Sum_{k=1..n} prime(k+1)*floor(n/prime(k+1)).

%C Sum of all odd prime divisors of all positive integers <= n.

%H <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>

%F G.f.: (1/(1 - x))*Sum_{k>=2} prime(k)*x^prime(k)/(1 - x^prime(k)).

%F a(n) = -2*floor(n/2) + Sum_{k=1..n} prime(k)*floor(n/prime(k)) .

%F a(n) = A024924(n) - A052928(n).

%e For n = 7 the odd prime divisors of the first seven positive integers are {0}, {0}, {3}, {0}, {5}, {3}, {7} so a(7) = 0 + 0 + 3 + 0 + 5 + 3 + 7 = 18.

%t Table[Sum[Prime[k + 1] Floor[n/Prime[k + 1]], {k, 1, n}], {n, 70}]

%t Rest[nmax = 70; CoefficientList[Series[(1/(1 - x)) Sum[Prime[k] x^Prime[k]/(1 - x^Prime[k]), {k, 2, nmax}], {x, 0, nmax}], x]]

%Y Cf. A005069, A008472, A024924, A052928.

%K nonn,easy

%O 1,3

%A _Ilya Gutkovskiy_, Dec 24 2016