A381978 a(n) is the smallest number k such that b+c+d = n, where b, c and d are three distinct positive divisors of k.

6, 4, 10, 6, 6, 6, 12, 8, 8, 12, 10, 10, 12, 12, 12, 12, 12, 14, 24, 16, 16, 18, 16, 18, 18, 20, 20, 18, 20, 20, 24, 24, 24, 24, 24, 26, 24, 28, 24, 30, 28, 30, 30, 28, 30, 30, 32, 34, 36, 30, 32, 36, 36, 38, 36, 40, 40, 36, 40, 40, 36, 44, 40, 42, 40, 46, 48, 48, 48, 48, 48, 42, 48, 52, 48, 54, 52, 54, 48, 50, 56, 54, 48, 58, 54, 52, 56, 60, 60, 60, 60, 64, 56, 54

Offset

6,1

Comments

a(n) may have more than three distinct positive divisors. The sequence is well-defined for n >= 6 since 2*(n-3) has divisors 1, 2 and n-3 whose sum is n.

The 2022 International Mathematical Olympiad asks for a(2022).

Links

David A. Corneth, Table of n, a(n) for n = 6..10000

International Mathematical Olympiad, Shortlisted Problems with Solutions, 2022. Problem N1, p7.

David A. Corneth, PARI program

Example

a(6) = 6 because 6=1+2+3 is the smallest solution.
a(7) = 4 because 7=1+2+4 is the smallest solution.

Prog

(Python)
from sympy import divisors
def a(n):
    for l in range(1, 3*n):
        divs = divisors(l)
        if len(divs) < 3:
            continue
        for i in range(len(divs)):
            for j in range(i+1, len(divs)):
                for m in range(j+1, len(divs)):
                    if divs[i] + divs[j] + divs[m] == n:
                        return l
    return None
print([a(n) for n in range(6, 100)])
(PARI) a(n)={for(m=1, 2*(n-3), my(d=divisors(m)); for(i=3, #d, for(j=2, i-1, for(k=1, j-1, if(d[i]+d[j]+d[k]==n, return(m)) )))); oo} \\ Andrew Howroyd, Mar 12 2025
(PARI) \\ See Corneth link

Crossrefs

Sequence in context: A143520 A075450 A145979 * A182164 A294093 A257926

Adjacent sequences: A381975 A381976 A381977 * A381979 A381980 A381981

Keyword

nonn

Author

Aksel Eide-Hansen, Mar 11 2025

Status

approved