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 1-5 of 5 results.

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

Views

Author

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

Keywords

Comments

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

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. A066353, A079645, A112873.

Programs

  • Magma
    [k*j: j in [(k^2+1)..(k^2+3*k+3)], k in [1..6]]; // 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 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
  • Mathematica
    Select[Range[300],!IntegerQ[Surd[#,3]]&&Divisible[#,Floor[Surd[#,3]]]&] (* Harvey P. Dale, May 13 2020 *)
  • Python
    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
  • SageMath
    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

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

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).

A073087 Least k such that sigma(k^k)>=n*k^k.

Original entry on oeis.org

1, 6, 30, 210, 30030, 223092870, 13082761331670030, 3217644767340672907899084554130, 1492182350939279320058875736615841068547583863326864530410
Offset: 1

Views

Author

Benoit Cloitre, Aug 18 2002

Keywords

Comments

Does a(n) = the product of primes less than or equal to prime(n+1) = A002110(n+1)? Answer from Lambert Klasen (Lambert.Klasen(AT)gmx.net), Sep 14 2005: No, this is not true.
Note that sigma(k^k) = prod (p^(k r + 1) - 1)/(p - 1). - Mitch Harris, Sep 14 2005
I have proved to my own satisfaction that for n >= 4, A073087(n) = p#, where p is the smallest prime satisfying p#/phi(p#) >= n. See link. - David W. Wilson, Sep 14 2005

Links

Crossrefs

Cf. A023199.

Programs

  • PARI
    a(n)=if(n<0,0,s=1; while(sigma(s^s)

Formula

a(n) = A091440(n)# = A002110(A112873(n)) for n >= 4.

Extensions

More terms from David W. Wilson, Sep 15 2005

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 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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