A107331 - OEIS

%I #19 Dec 10 2025 17:00:45

%S 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,

%T 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,

%U 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

%N 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.

%C 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.

%H Aloe Poliszuk, <a href="/A107331/b107331.txt">Table of n, a(n) for n = 1..10000</a>

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

%t 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 *)

%o (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

%Y Cf. A033635, A086671, A058266.

%K nonn,mult

%O 1,2

%A _Yasutoshi Kohmoto_, May 23 2005

%E Edited, corrected and extended by _Robert G. Wilson v_, Jun 08 2005