A273777 Consider all ways of writing the n-th composite number as the product of two divisors d1*d2 = d3*d4 = ... where each divisor is larger than 1; a(n) is the maximum of the sums {d1 + d2, d3 + d4, ...}.

4, 5, 6, 6, 7, 8, 9, 8, 10, 11, 12, 10, 13, 14, 10, 15, 12, 16, 17, 18, 14, 19, 12, 20, 21, 16, 22, 23, 24, 18, 25, 26, 14, 27, 20, 28, 29, 16, 30, 22, 31, 32, 33, 24, 34, 18, 35, 36, 26, 37, 38, 39, 28, 40, 18, 41, 42, 30, 43, 44, 22, 45, 32, 46, 47, 20, 48

Offset

1,1

Comments

The divisors must be > 1 and < n.

For the minimum sums see A273227.

Links

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Formula

Let m = A002808(n). Then a(n) = A020639(m) + m / A020639(m).

Example

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

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:
    n0:=nops(lst):printf(`%d, `, lst[n0]):
   fi:
   od:

Mathematica

Function[n, Max@ Map[Plus[#, n/#] &, Rest@ Take[#, Ceiling[Length[#]/2]]] &@ Divisors@ n] /@ Select[Range@ 120, CompositeQ] (* Michael De Vlieger, May 30 2016 *)

Prog

(PARI) lista(nn) = {forcomposite(n=2, nn, m = 0; fordiv(n, d, if ((d != 1) && (d != n), m = max(m, d+n/d)); ); print1(m, ", "); ); } \\ Michel Marcus, Sep 13 2017
(Python)
from sympy import composite, divisors
def A273777(n): return (d:=divisors(composite(n)))[1] + d[-2] # Karl-Heinz Hofmann, Nov 16 2025
(Python)
from sympy import divisors
def first(n):
    a, k = [-1], 4
    while len(a) <= n:
        a.append(2 + k//2)
        if len(d:=divisors(k + 1)) > 2: a.append(d[1] + d[-2])
        k += 2
    return a[1:n+1]
print(first(100)) # Karl-Heinz Hofmann, Nov 16 2025

Crossrefs

Cf. A002808, A020639, A046343, A063655, A273227.

Sequence in context: A022911 A162310 A116962 * A023846 A338625 A046345

Adjacent sequences: A273774 A273775 A273776 * A273778 A273779 A273780

Keyword

nonn

Author

Michel Lagneau, May 30 2016

Extensions

Name edited by Jon E. Schoenfield, Sep 12 2017

Status

approved