A273227 Consider all ways of writing the n-th composite number as the product of two divisors d1*d2 = d3*d4 = ...; a(n) is the minimum of the sums {d1 + d2, d3 + d4, ...}.

4, 5, 6, 6, 7, 7, 9, 8, 8, 9, 9, 10, 13, 10, 10, 15, 12, 11, 11, 12, 14, 19, 12, 12, 21, 16, 13, 13, 15, 14, 25, 14, 14, 15, 20, 17, 15, 16, 15, 22, 31, 16, 33, 16, 16, 18, 17, 21, 26, 17, 17, 39, 20, 23, 18, 19, 18, 18, 43, 19, 22, 45, 32, 19, 19, 20, 27, 34

Offset: 1

Author

Michel Lagneau, May 18 2016

Keywords

• nonn

Comments

a(n) = A046343(n) if n is semiprime.

This sequence is included in A063655. - Giovanni Resta, May 18 2016

a(n) >= 2 * sqrt(A002808(n)). - David A. Corneth, May 20 2016

Examples

			a(14) = 10 because A002808(14) = 24 = 2*12 = 3*8 = 4*6 and 4+6 = 10 is the minimum sum.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002808, A046343, A063655.

Programs

Maple

with(numtheory):nn:=100:lst:={}:
for n from 1 to nn do:
it:=0:lst:={}:
d:=divisors(n):n0:=nops(d):
  if n0>2 then
  for i from 2 to n0-1 do:
   p:=d[i]:
    for j from i to n0-1 do:
      q:=d[j]:
       if p*q=n then
        lst:=lst union {p+q}:
        else
       fi:
     od:
    od:
    printf(`%d, `,lst[1]):
   fi:
   od:

Mathematica

Function[n, If[OddQ@ Length@ #, 2 Sqrt@ n, Total@ Take[#, {Length[#]/2, Length[#]/2 + 1}]] &@ Divisors@ n] /@ Select[Range@ 93, CompositeQ] (* Michael De Vlieger, May 20 2016 *)
msd[n_]:=Module[{d=Divisors[n],len},len=Length[d];If[OddQ[len], 2*d[[ (len+1)/2]], d[[len/2]]+d[[len/2+1]]]]; msd/@Select[Range[200], CompositeQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 09 2018 *)

PARI

forcomposite(n=4,100, d=divisors(n); t=#d; k=if(t%2,2*d[t\2+1], d[t\2]+d[t\2+1]); print1(k", ")) \\ Charles R Greathouse IV, Jun 08 2016

Extensions

Name edited by Jon E. Schoenfield, Sep 12 2017