cp's OEIS Frontend

A365409 a(n) = Sum_{k=1..n} binomial(floor(n/k)+3,4).

%I A365409 #25 Oct 27 2023 09:57:13
%S A365409 1,6,17,42,78,149,234,379,555,815,1102,1557,2013,2662,3388,4349,5319,
%T A365409 6695,8026,9846,11712,14027,16328,19503,22464,26200,30030,34759,39255,
%U A365409 45221,50678,57623,64465,72579,80469,90665,99805,111020,122146,135566,147908,163638
%N A365409 a(n) = Sum_{k=1..n} binomial(floor(n/k)+3,4).
%F A365409 a(n) = Sum_{k=1..n} binomial(k+2,3) * floor(n/k).
%F A365409 G.f.: 1/(1-x) * Sum_{k>=1} x^k/(1-x^k)^4 = 1/(1-x) * Sum_{k>=1} binomial(k+2,3) * x^k/(1-x^k).
%F A365409 a(n) = (A064603(n)+3*A064602(n)+2*A024916(n))/6. - _Chai Wah Wu_, Oct 26 2023
%o A365409 (PARI) a(n) = sum(k=1, n, binomial(n\k+3, 4));
%o A365409 (Python)
%o A365409 from math import isqrt, comb
%o A365409 def A365409(n): return -(s:=isqrt(n))**2*comb(s+3,3)+sum((q:=n//k)*((comb(k+2,3)<<2)+comb(q+3,3)) for k in range(1,s+1))>>2 # _Chai Wah Wu_, Oct 26 2023
%Y A365409 Partial sums of A059358.
%Y A365409 Cf. A006218, A024916, A064602, A064603, A364970, A365439.
%K A365409 nonn,easy
%O A365409 1,2
%A A365409 _Seiichi Manyama_, Oct 23 2023