A362739 The smallest integer with three (not necessarily distinct) divisors that add to n.

1, 2, 2, 2, 3, 4, 3, 4, 5, 4, 6, 6, 5, 8, 8, 6, 9, 8, 7, 10, 11, 8, 10, 12, 9, 12, 14, 10, 15, 16, 11, 16, 14, 12, 18, 18, 13, 16, 20, 14, 21, 20, 15, 22, 23, 16, 21, 20, 17, 24, 26, 18, 22, 24, 19, 28, 29, 20, 30

Offset

3,2

Comments

a(n) is bounded between n/3 and n/2.

The smallest integer with 2 (instead of 3) divisors that sum to n is given by A060681.

Links

Table of n, a(n) for n=3..61.

Formula

a(2^k) = 2^(k-1).

If n is not a power of 2, a(n) = n*(p-1)/(2p), where p is the smallest odd prime dividing n.

Example

a(6) = 2 as one can have 2+2+2 = 6.
a(55) = 22 as one can have 11+22+22 = 55.

Maple

f:= proc(n) local P, p;
  P:= numtheory:-factorset(n) minus {2};
  if P = {} then return n/2 fi;
  p:= min(P);
  n*(p-1)/(2*p)
end proc:
map(f, [$3..100]); # Robert Israel, May 01 2023

Prog

(SageMath)
def a(n):
    F = list(factor(n))
    if F[0][0] == 2:
        if len(F) == 1:
            return n / 2
        del F[0]
    m = F[0][0]
    return n * (m - 1) / (2 * m)
(Python)
from sympy import primefactors
def A362739(n): return n>>1 if (m:=n&-n)==n else n*((p:=min(primefactors(n//m)))-1)//p>>1 # Chai Wah Wu, Jun 21 2023

Crossrefs

Cf. A060681, A020639.

Sequence in context: A186963 A060473 A055034 * A112184 A112213 A238957

Adjacent sequences: A362736 A362737 A362738 * A362740 A362741 A362742

Keyword

nonn,easy

Author

Kyan Cheung, May 01 2023

Status

approved