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A112873 Partial sums of A032378.

%I A112873 #17 Oct 13 2024 03:45:23
%S A112873 2,5,9,14,20,27,37,49,63,79,97,117,139,163,189,219,252,288,327,369,
%T A112873 414,462,513,567,624,684,747,815,887,963,1043,1127,1215,1307,1403,
%U A112873 1503,1607,1715,1827,1943,2063,2187,2317,2452,2592,2737,2887,3042,3202,3367,3537
%N A112873 Partial sums of A032378.
%H A112873 G. C. Greubel, <a href="/A112873/b112873.txt">Table of n, a(n) for n = 1..10000</a>
%F A112873 From _Vaclav Kotesovec_, Oct 13 2024: (Start)
%F A112873 a(3*k*(k+3)/2) = 3*k*(k+1)*(k+2)*(8*k^2+21*k+31)/40.
%F A112873 a(n) ~ 2^(5/2)*n^(5/2)/(5*3^(3/2)) - n^2/2 + 13*n^(3/2)/(2^(3/2)*sqrt(3)). (End)
%t A112873 Accumulate[Select[Range[300],!IntegerQ[Surd[#,3]]&&Divisible[#,Floor[ Surd[ #,3]]]&]] (* _Harvey P. Dale_, May 13 2020 *)
%o A112873 (Magma)
%o A112873 A032378:=[k*j: j in [(k^2+1)..(k^2+3*k+3)], k in [1..15]];
%o A112873 [(&+[A032378[j]: j in [1..n]]): n in [1..100]]; // _G. C. Greubel_, Jul 20 2023
%o A112873 (SageMath)
%o A112873 A032378=flatten([[k*j for j in range((k^2+1),(k^2+3*k+3)+1)] for k in range(1,15)])
%o A112873 def A112873(n): return sum(A032378[j] for j in range(n+1))
%o A112873 [A112873(n) for n in range(101)] # _G. C. Greubel_, Jul 20 2023
%o A112873 (Python)
%o A112873 from itertools import count, islice, accumulate
%o A112873 from sympy import integer_nthroot
%o A112873 def A112873_gen(): # generator of terms
%o A112873     return accumulate(filter(lambda x: not x%integer_nthroot(x,3)[0],(n+(k:=integer_nthroot(n, 3)[0])+int(n>=(k+1)**3-k) for n in count(1))))
%o A112873 A112873_list = list(islice(A112873_gen(),40)) # _Chai Wah Wu_, Oct 12 2024
%Y A112873 Cf. A032378, A066353, A120721.
%K A112873 nonn
%O A112873 1,1
%A A112873 _N. J. A. Sloane_, Oct 29 2006