A066353 -id:A066353 - cp's OEIS frontend

A112873 Partial sums of A032378.

2, 5, 9, 14, 20, 27, 37, 49, 63, 79, 97, 117, 139, 163, 189, 219, 252, 288, 327, 369, 414, 462, 513, 567, 624, 684, 747, 815, 887, 963, 1043, 1127, 1215, 1307, 1403, 1503, 1607, 1715, 1827, 1943, 2063, 2187, 2317, 2452, 2592, 2737, 2887, 3042, 3202, 3367, 3537

Offset: 1

N. J. A. Sloane, Oct 29 2006

nonn

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A032378, A066353, A120721.

A032378:=[k*j: j in [(k^2+1)..(k^2+3*k+3)], k in [1..15]];
[(&+[A032378[j]: j in [1..n]]): n in [1..100]]; // G. C. Greubel, Jul 20 2023

Accumulate[Select[Range[300],!IntegerQ[Surd[#,3]]&&Divisible[#,Floor[ Surd[ #,3]]]&]] (* Harvey P. Dale, May 13 2020 *)

from itertools import count, islice, accumulate
from sympy import integer_nthroot
def A112873_gen(): # generator of terms
    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))))
A112873_list = list(islice(A112873_gen(),40)) # Chai Wah Wu, Oct 12 2024

A032378=flatten([[k*j for j in range((k^2+1),(k^2+3*k+3)+1)] for k in range(1,15)])
def A112873(n): return sum(A032378[j] for j in range(n+1))
[A112873(n) for n in range(101)] # G. C. Greubel, Jul 20 2023

From Vaclav Kotesovec, Oct 13 2024: (Start)
a(3*k*(k+3)/2) = 3*k*(k+1)*(k+2)*(8*k^2+21*k+31)/40.
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)

A032378 Noncubes k that are divisible by floor(k^(1/3)).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 10, 12, 14, 16, 18, 20, 22, 24, 26, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 222, 228, 234, 240

Offset: 1

N. J. A. Sloane, Dec 22 2001, corrected Oct 29 2006

nonn

The Concrete Math Club Casino problem - non-cube winning slots.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. 2nd Edition. Addison-Wesley, Reading, MA, 1994. Section 3.2, pp. 74-76.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A066353, A079645, A112873.

[k*j: j in [(k^2+1)..(k^2+3*k+3)], k in [1..6]]; // G. C. Greubel, Jul 20 2023

t1:=[]; for n from 1 to 500 do t2:=floor(n^(1/3)); if n mod t2 = 0 and t2^3 <> n then t1:=[op(t1),n]; fi; od:
# Alternate:
seq(seq(n,n=k^3+k..(k+1)^3-1,k),k=1..6); # Robert Israel, Mar 24 2020

Select[Range[300],!IntegerQ[Surd[#,3]]&&Divisible[#,Floor[Surd[#,3]]]&] (* Harvey P. Dale, May 13 2020 *)

from itertools import count, islice
from sympy import integer_nthroot
def A032378_gen(): # generator of terms
    return 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)))
A032378_list = list(islice(A032378_gen(),40)) # Chai Wah Wu, Oct 12 2024

flatten([[k*j for j in range((k^2+1),(k^2+3*k+3)+1)] for k in range(1,7)]) # G. C. Greubel, Jul 20 2023

cp's OEIS Frontend

A112873 Partial sums of A032378.

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