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.

A086671 Sum of floor(sqrt(d)) where d runs through the divisors of n.

Original entry on oeis.org

1, 2, 2, 4, 3, 5, 3, 6, 5, 7, 4, 10, 4, 7, 7, 10, 5, 12, 5, 13, 8, 9, 5, 16, 8, 10, 10, 14, 6, 18, 6, 15, 10, 11, 10, 23, 7, 12, 11, 21, 7, 20, 7, 17, 16, 12, 7, 26, 10, 19, 13, 19, 8, 24, 13, 23, 13, 14, 8, 34, 8, 14, 18, 23, 14, 25, 9, 21, 14, 25, 9, 37, 9
Offset: 1

Views

Author

Jon Perry, Jul 27 2003

Keywords

Examples

			10 has divisors 1,2,5,10. floor(sqrt(d)) gives 1,1,2,3, therefore a(10)=7.

Links

Crossrefs

Cf. A332931, A332932, A332933, A332934, A332935.
Cf. A135539.

Programs

  • Maple
    A086671:= proc(n)
    add(floor(sqrt(d)), d = numtheory[divisors](n))
end proc; # R. J. Mathar, Oct 26 2013
  • Mathematica
    Table[DivisorSum[n, Floor[Sqrt[#]] &], {n, 100}] (* T. D. Noe, Oct 28 2013 *)
  • PARI
    for (n=1,100,s=0; fordiv(i=n,i,s+=floor(sqrt(i))); print1(","s))
  • PARI
    a(n) = sumdiv(n, d, sqrtint(d)); \\ Michel Marcus, Mar 03 2020

Formula

a(n) = Sum_{d|n} floor(sqrt(d)). - Wesley Ivan Hurt, Oct 25 2013
G.f.: sum(k>=1, floor(sqrt(k))*x^k/(1-x^k) ). - Mircea Merca, Feb 22 2014
a(n) = Sum_{i=1..floor(sqrt(n))} A135539(n,i^2). - Ridouane Oudra, Apr 15 2022

A332932 Sum of ceiling(sqrt(d)) where d runs through the divisors of n.

Original entry on oeis.org

1, 3, 3, 5, 4, 8, 4, 8, 6, 10, 5, 14, 5, 10, 10, 12, 6, 16, 6, 17, 11, 12, 6, 22, 9, 13, 12, 18, 7, 25, 7, 18, 13, 14, 13, 28, 8, 15, 14, 27, 8, 27, 8, 21, 20, 15, 8, 33, 11, 23, 16, 23, 9, 30, 16, 29, 16, 17, 9, 44, 9, 17, 22, 26, 17, 32, 10, 25, 17, 32, 10
Offset: 1

Views

Author

Harvey P. Dale, Mar 02 2020

Keywords

Links

Crossrefs

Cf. A058268, A086671, A332931, A332933, A332934, A332935.

Programs

  • Maple
    a:= n-> add(ceil(sqrt(d)), d=numtheory[divisors](n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Mar 02 2020
  • Mathematica
    Table[DivisorSum[n,Ceiling[Sqrt[#]]&],{n,80}]
  • PARI
    a(n) = sumdiv(n, d, ceil(sqrt(d))); \\ Michel Marcus, Mar 03 2020
  • Python
    from math import isqrt
from sympy import divisors
def A332932(n): return sum(1+isqrt(d-1) for d in divisors(n,generator=True)) # Chai Wah Wu, Jul 28 2022

A332933 Sum of floor(d^(3/2)) where d runs through the divisors of n.

Original entry on oeis.org

1, 3, 6, 11, 12, 22, 19, 33, 33, 45, 37, 71, 47, 73, 75, 97, 71, 125, 83, 142, 120, 142, 111, 210, 137, 181, 173, 229, 157, 286, 173, 278, 231, 271, 237, 390, 226, 319, 295, 416, 263, 460, 282, 441, 403, 424, 323, 606, 362, 523, 440, 563, 386, 661, 455, 670
Offset: 1

Views

Author

Harvey P. Dale, Mar 02 2020

Keywords

Links

Crossrefs

Cf. A058269, A086671, A332931, A332932, A332934, A332935.

Programs

  • Maple
    a:= n-> add(floor(d^(3/2)), d=numtheory[divisors](n)):
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 02 2020
  • Mathematica
    Table[DivisorSum[n,Floor[Sqrt[#^3]]&],{n,80}]
  • PARI
    a(n)={sumdiv(n, d, sqrtint(d^3))} \\ Andrew Howroyd, Mar 02 2020

A332934 Sum of round(d^(3/2)) where d runs through the divisors of n.

Original entry on oeis.org

1, 4, 6, 12, 12, 24, 20, 35, 33, 47, 37, 74, 48, 75, 75, 99, 71, 127, 84, 144, 121, 143, 111, 215, 137, 184, 173, 231, 157, 289, 174, 280, 232, 272, 238, 393, 226, 321, 297, 420, 264, 463, 283, 443, 404, 426, 323, 612, 363, 526, 440, 567, 387, 664, 456, 673
Offset: 1

Views

Author

Harvey P. Dale, Mar 02 2020

Keywords

Links

Crossrefs

Cf. A058270, A086671, A332931, A332932, A332933, A332935.

Programs

  • Maple
    a:= n-> add(round(d^(3/2)), d=numtheory[divisors](n)):
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 02 2020
  • Mathematica
    Table[DivisorSum[n,Floor[1/2+Sqrt[#^3]]&],{n,80}]
  • Python
    from math import isqrt
from sympy import divisors
def A332934(n): return sum((m:=isqrt(r:=d**3))+int(r-m*(m+1)>=1) for d in divisors(n,generator=True)) # Chai Wah Wu, Aug 03 2022

A332935 Sum of ceiling(n^(3/2)) where d runs through the divisors of n.

Original entry on oeis.org

1, 4, 7, 12, 13, 25, 20, 35, 34, 48, 38, 75, 48, 76, 78, 99, 72, 129, 84, 146, 123, 145, 112, 216, 138, 184, 175, 233, 158, 293, 174, 281, 234, 274, 240, 395, 227, 322, 298, 422, 264, 467, 283, 445, 407, 427, 324, 613, 363, 527, 443, 567, 387, 667, 458, 676
Offset: 1

Views

Author

Harvey P. Dale, Mar 02 2020

Keywords

Links

Crossrefs

Cf. A058271, A086671, A332931, A332932, A332933, A332934.

Programs

  • Maple
    a:= n-> add(ceil(d^(3/2)), d=numtheory[divisors](n)):
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 02 2020
  • Mathematica
    Table[DivisorSum[n,Ceiling[Sqrt[#^3]]&],{n,80}]
  • PARI
    a(n)={sumdiv(n, d, 1 + sqrtint(d^3 - 1))} \\ Andrew Howroyd, Mar 02 2020
  • Python
    from math import isqrt
from sympy import divisors
def A332935(n): return sum(1+isqrt(d**3-1) for d in divisors(n,generator=True)) # Chai Wah Wu, Aug 03 2022
Showing 1-5 of 5 results.

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