cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A112873 Partial sums of A032378.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Oct 29 2006

Keywords

Links

Crossrefs

Cf. A032378, A066353, A120721.

Programs

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

Formula

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)

A066353 1 + partial sums of A032378.

Original entry on oeis.org

1, 3, 6, 10, 15, 21, 28, 38, 50, 64, 80, 98, 118, 140, 164, 190, 220, 253, 289, 328, 370, 415, 463, 514, 568, 625, 685, 748, 816, 888, 964, 1044, 1128, 1216, 1308, 1404, 1504, 1608, 1716, 1828, 1944, 2064, 2188, 2318, 2453, 2593, 2738, 2888
Offset: 0

Views

Author

N. J. A. Sloane, Dec 22 2001

Keywords

Comments

A032378 has been inspired by the Concrete Mathematics Casino problem (see reference).

References

  • R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. 2nd Edition. Addison-Wesley, Reading, MA, 1994. Section 3.2, p74-76.

Links

Crossrefs

Cf. A032378, A112873.

Programs

  • Magma
    A032378:=[k*j: j in [(k^2+1)..(k^2+3*k+3)], k in [1..15]];
[n eq 0 select 1 else 1+(&+[A032378[j]: j in [1..n]]): n in [0..100]]; // G. C. Greubel, Jul 20 2023
  • Mathematica
    A032378:= A032378= Table[k*j, {k,15}, {j,k^2+1, k^2+3*k+3}]//Flatten;
A066353[n_]:= A066353[n]= 1 +Sum[A032378[[j+1]], {j,0,n-1}];
Table[A066353[n], {n,0,100}] (* G. C. Greubel, Jul 20 2023 *)
Accumulate[Join[{1},Select[Range[300],!IntegerQ[Surd[#,3]]&&Divisible[#,Floor[Surd[#,3]]]&]]] (* Harvey P. Dale, Jan 25 2025 *)
  • SageMath
    A032378=flatten([[k*j for j in range((k^2+1),(k^2+3*k+3)+1)] for k in range(1,15)])
def A066353(n): return 1 if (n==0) else 1 + sum(A032378[j] for j in range(n))
[A066353(n) for n in range(101)] # G. C. Greubel, Jul 20 2023

Formula

a(n) = 1 if n = 0, otherwise a(n) = A112873(n) = Sum_{j=1..n} A032378(j).

A079645 Numbers j such that the integer part of the cube root of j divides j.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210
Offset: 1

Views

Author

Benoit Cloitre, Jan 31 2003

Keywords

Comments

Concrete Mathematics Casino Problem - Winners.

Examples

			252^(1/3) = 6.316359597656... and 252/6 = 42 hence 252 is in the sequence.

References

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

Links

Crossrefs

Cf. A006446, A032378, A079631, A112873, A120721.

Programs

  • Magma
    [n: n in [1..250] | n mod Floor(n^(1/3)) eq 0 ]; // G. C. Greubel, Jul 20 2023
  • Maple
    t1:=[]; for n from 1 to 500 do t2:=floor(n^(1/3)); if n mod t2 = 0 then t1:=[op(t1),n]; fi; od: t1; # N. J. A. Sloane, Oct 29 2006
  • Mathematica
    Select[Range[1000], Mod[#, Floor[Power[#, 1/3]]] == 0 &]
Select[Range[1000],Divisible[#,Floor[CubeRoot[#]]]&] (* Harvey P. Dale, Jun 19 2023 *)
  • SageMath
    [n for n in (1..250) if n%(floor(n^(1/3)))==0 ] # G. C. Greubel, Jul 20 2023

Formula

For n = (k/2)*(3*k+11) - m for some fixed m >= 0 with n > ((k-1)/2)*(3*(k-1) + 11) we have a(n) = k^3 + 3*k^2 + (3-m)*k. - Benoit Cloitre, Jan 22 2012

A120721 Partial sums of A079645.

Original entry on oeis.org

1, 3, 6, 10, 15, 21, 28, 36, 46, 58, 72, 88, 106, 126, 148, 172, 198, 225, 255, 288, 324, 363, 405, 450, 498, 549, 603, 660, 720, 783, 847, 915, 987, 1063, 1143, 1227, 1315, 1407, 1503, 1603, 1707, 1815, 1927, 2043, 2163, 2287, 2412, 2542, 2677, 2817, 2962, 3112, 3267
Offset: 1

Views

Author

N. J. A. Sloane, Oct 29 2006

Keywords

Links

Crossrefs

Cf. A079645, A032378, A112873.

Programs

  • Magma
    A079645:=[n: n in [1..500] | n mod Floor(n^(1/3)) eq 0 ];
[(&+[A079645[k]: k in [1..n]]): n in [1..100]]; // G. C. Greubel, Jul 20 2023
  • Mathematica
    Accumulate[Select[Range[300],Divisible[#,Floor[CubeRoot[#]]]&]] (* Harvey P. Dale, Jun 19 2023 *)
  • SageMath
    A079645=[j for j in (1..500) if j%(floor(j^(1/3)))==0]
def A120721(n): return sum(A079645[k] for k in range(n+1))
[A120721(n) for n in range(101)] # G. C. Greubel, Jul 20 2023

Formula

a(n) = Sum_{j=1..n} A079645(j).
a(n) ~ 2^(5/2)*n^(5/2)/(5*3^(3/2)) - 5*n^2/6. - Vaclav Kotesovec, Oct 13 2024
Showing 1-4 of 4 results.

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