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.

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

1,2

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.

Links

Aloe Poliszuk, Table of n, a(n) for n = 1..10000

Example

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

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 *)

Prog

(PARI) a(n)=my(f=factorint(n)); prod(i=1, #f~, floor((f[i, 1]^(f[i, 2]/2+1/2)-1)/(f[i, 1]^(1/2)-1))); \\ Aloe Poliszuk, Dec 09 2025

Crossrefs

Cf. A033635, A086671, A058266.

Sequence in context: A274176 A356148 A083742 * A283187 A324391 A357978

Adjacent sequences: A107328 A107329 A107330 * A107332 A107333 A107334

Keyword

nonn,mult

Author

Yasutoshi Kohmoto, May 23 2005

Extensions

Edited, corrected and extended by Robert G. Wilson v, Jun 08 2005

Status

approved