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 1-8 of 8 results.

A332931 Sum of round(sqrt(d)) where d runs through the divisors of n.

Original entry on oeis.org

1, 2, 3, 4, 3, 6, 4, 7, 6, 7, 4, 11, 5, 9, 9, 11, 5, 13, 5, 13, 11, 10, 6, 19, 8, 11, 11, 16, 6, 20, 7, 17, 12, 12, 12, 24, 7, 12, 13, 22, 7, 24, 8, 19, 19, 14, 8, 30, 11, 19, 14, 20, 8, 25, 13, 26, 15, 15, 9, 37, 9, 16, 22, 25, 15, 28, 9, 22, 16, 28, 9, 40
Offset: 1

Views

Author

Harvey P. Dale, Mar 02 2020

Keywords

Links

Crossrefs

Cf. A058267 (which has the "round" outside the sum), A086671, A332932, A332933, A332934, A332935.

Programs

  • Maple
    a:= n-> add(round(sqrt(d)), d=numtheory[divisors](n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Mar 02 2020
  • Mathematica
    Table[DivisorSum[n,Floor[1/2+Sqrt[#]]&],{n,80}]
  • PARI
    a(n) = sumdiv(n, d, round(sqrt(d))); \\ Michel Marcus, Mar 03 2020
  • Python
    from math import isqrt
from sympy import divisors
def A332931(n): return sum((m:=isqrt(d))+int(d-m*(m+1)>=1) for d in divisors(n,generator=True)) # Chai Wah Wu, Aug 03 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

A058266 An approximation to sigma_{1/2}(n): floor( sum_{ d divides n } sqrt(d) ).

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 3, 7, 5, 7, 4, 12, 4, 8, 8, 11, 5, 13, 5, 14, 9, 10, 5, 19, 8, 11, 10, 16, 6, 21, 6, 16, 11, 12, 11, 25, 7, 12, 12, 23, 7, 24, 7, 19, 18, 13, 7, 30, 10, 19, 13, 20, 8, 26, 13, 26, 14, 15, 8, 39, 8, 15, 20, 24, 14, 28, 9, 22, 15, 28, 9, 41, 9
Offset: 1

Views

Author

N. J. A. Sloane, Dec 08 2000

Keywords

Links

Crossrefs

Cf. A000203, A001157, A058267, A058268, A078434, A086671.

Programs

  • Maple
    with(numtheory); f := proc(n) local d, t1, t2; t2 := 0; t1 := divisors(n); for d in t1 do t2 := t2 + sqrt(d) end do; t2 end proc; # exact value of sigma_{1/2}(n)
with(numtheory):seq(floor(sigma[1/2](n)),n=1..80);
  • Mathematica
    f[n_] := Floor@DivisorSigma[1/2, n]; Array[f, 73] (* Robert G. Wilson v, Aug 17 2017*)
  • PARI
    a(n) = floor(sumdiv(n, d, sqrt(d))); \\ Michel Marcus, Aug 17 2017

Formula

Sum_{k=1..n} a(k) ~ (2/3)*zeta(3/2) * n^(3/2). - Amiram Eldar, Jan 14 2023

A344460 a(n) = Sum_{d|n} d * floor(sqrt(d)).

Original entry on oeis.org

1, 3, 4, 11, 11, 18, 15, 27, 31, 43, 34, 62, 40, 59, 59, 91, 69, 117, 77, 131, 102, 124, 93, 174, 136, 172, 166, 207, 146, 253, 156, 251, 202, 241, 200, 377, 223, 307, 277, 387, 247, 410, 259, 396, 356, 371, 283, 526, 358, 518, 429, 544, 372, 630, 429, 615, 479, 554, 414, 797
Offset: 1

Views

Author

Wesley Ivan Hurt, May 19 2021

Keywords

Comments

Inverse Möbius transform of n * floor(sqrt(n)). - Wesley Ivan Hurt, Mar 31 2025

Examples

			a(10) = Sum_{d|10} d * floor(sqrt(d)) = 1*1 + 2*1 + 5*2 + 10*3 = 43.

Crossrefs

Cf. A000196, A086671.

Programs

  • Mathematica
    Table[Sum[k*Floor[Sqrt[k]] (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 80}]

A107331 An approximation to sigma_{1/2}(n): multiplicative with a(p^e) = floor((p^(e/2+1/2)-1)/(p^(1/2)-1)) for prime p.

Original entry on oeis.org

1, 2, 2, 4, 3, 4, 3, 7, 5, 6, 4, 8, 4, 6, 6, 11, 5, 10, 5, 12, 6, 8, 5, 14, 8, 8, 10, 12, 6, 12, 6, 16, 8, 10, 9, 20, 7, 10, 8, 21, 7, 12, 7, 16, 15, 10, 7, 22, 10, 16, 10, 16, 8, 20, 12, 21, 10, 12, 8, 24, 8, 12, 15, 24, 12, 16, 9, 20, 10, 18, 9, 35, 9, 14, 16, 20, 12, 16, 9, 33, 19, 14
Offset: 1

Views

Author

Yasutoshi Kohmoto, May 23 2005

Keywords

Comments

Whereas A086671 takes the sum of the floor of the square roots of each of the divisors of n and A058266 takes the floor of the product formula, this sequence takes the product of the floor of the individual prime components of the product formula.

Examples

			a(8) = floor((2^((3+1)/2)-1)/(2^(1/2)-1)) = floor(3/(sqrt(2)-1)) = floor(7.242...) = 7.

Crossrefs

Cf. A033635, A086671, A058266.

Programs

  • Mathematica
    f[n_] := Block[{pfe = FactorInteger[n]}, Times @@ Floor[((First /@ pfe)^((Last /@ pfe + 1)/2) - 1)/((First /@ pfe)^(1/2) - 1)]]; Table[ f[n], {n, 82}] (* Robert G. Wilson v, Jun 08 2005 *)

Extensions

Edited, corrected and extended by Robert G. Wilson v, Jun 08 2005
Showing 1-8 of 8 results.

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

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