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 1-3 of 3 results.

A058267 An approximation to sigma_{1/2}(n): round( Sum_{ d divides n } sqrt(d) ).

Original entry on oeis.org

1, 2, 3, 4, 3, 7, 4, 7, 6, 8, 4, 12, 5, 9, 9, 11, 5, 14, 5, 14, 10, 10, 6, 20, 8, 11, 11, 16, 6, 21, 7, 17, 12, 12, 12, 25, 7, 13, 13, 23, 7, 24, 8, 19, 19, 14, 8, 31, 11, 20, 14, 20, 8, 26, 14, 26, 15, 15, 9, 39, 9, 16, 21, 25, 15, 28, 9, 23, 16, 28, 9, 42, 10
Offset: 1

Views

Author

N. J. A. Sloane, Dec 08 2000

Keywords

Links

Crossrefs

Cf. A000203, A001157, A058266, A058268, A078434.

Programs

  • Maple
    map(round @ numtheory:-sigma[1/2], [$1..100]); # Robert Israel, Aug 18 2017
  • Mathematica
    f[n_] := Round@ DivisorSigma[1/2, n]; Array[f, 70] (* Robert G. Wilson v, Aug 17 2017 *)
  • PARI
    a(n) = round(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

A058268 An approximation to sigma_{1/2}(n): ceiling( sum_{d|n} sqrt(d) ).

Original entry on oeis.org

1, 3, 3, 5, 4, 7, 4, 8, 6, 8, 5, 13, 5, 9, 9, 12, 6, 14, 6, 15, 10, 11, 6, 20, 9, 12, 11, 17, 7, 22, 7, 17, 12, 13, 12, 26, 8, 13, 13, 24, 8, 25, 8, 20, 19, 14, 8, 31, 11, 20, 14, 21, 9, 27, 14, 27, 15, 16, 9, 40, 9, 16, 21, 25, 15, 29, 10, 23, 16, 29, 10, 42
Offset: 1

Views

Author

N. J. A. Sloane, Dec 08 2000

Keywords

Links

Crossrefs

Cf. A000203, A001157, A058266, A058267, A078434.

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)
  • Mathematica
    a[n_] := Ceiling[DivisorSigma[1/2, n]]; Array[a, 70] (* Amiram Eldar, Jan 14 2023 *)

Formula

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

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 1-3 of 3 results.

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

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