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A362739
The smallest integer with three (not necessarily distinct) divisors that add to n.
1
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.
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
Sequence in context: A186963 A060473 A055034 * A112184 A112213 A238957
KEYWORD
nonn,easy
AUTHOR
Kyan Cheung, May 01 2023
STATUS
approved