A346971 Smallest c which can be split into positive parts a and b with a+b=c, such that the divisors of a,b,c cover all numbers up to n.

2, 3, 4, 8, 10, 12, 24, 45, 54, 88, 120, 182, 182, 360, 540, 1326, 1326, 3990, 5040, 5040, 5040, 9282, 9282, 25200, 25200, 65208, 65208, 118800, 118800, 651456, 651456, 651456, 651456, 651456, 651456, 2314200, 2314200, 2314200, 2314200, 16365396, 16365396

Offset: 2

Author

Steven M. Altschuld, Aug 09 2021

Keywords

• nonn
• more

Comments

a(37)..a(40) <= 2314200 via 1062048 + 1252152 = 2314200. - David A. Corneth, Aug 11 2021

Examples

			a(5) = 8, 3+5=8, divisors of 3, 5, and 8 are {1,3}, {1,5}, and {1,2,4,8}, which covers all of {1,2,3,4,5}.
a(9) = 45, 21+24=45, divisors of 21, 24, and 45 are {1,3,7,21}, {1,2,3,4,6,8,12,24}, and {1,3,5,9,15,45}, which covers all of {1,2,3,4,5,6,7,8,9}.
		

Crossrefs

Cf. A003418, A027750, A346970.

Programs

Mathematica

a[1]=1;a[n_]:=(k=1;While[Length@Select[Union@*Flatten@*Divisors/@(Join[{k},#]&/@Rest@IntegerPartitions[k,2]),SubsetQ[#,Range@n]&]<1,k++];k);Array[a,16] (* Giorgos Kalogeropoulos, Aug 13 2021 *)

Python

from sympy import divisors
from itertools import count
def cond(a, b, c, n):
    return set(divisors(a)+divisors(b)+divisors(c)) >= set(range(1, n+1))
def a(n):
    if n == 1: return 1
    for c in count(1):
        for a in range(1, c//2+1):
            if cond(a, c-a, c, n): return c
print([a(n) for n in range(1, 17)]) # Michael S. Branicky, Aug 13 2021

Python

def A346971(n):
    c, nlist = 1, list(range(1,n+1))
    while True:
        mlist = [m for m in nlist if c % m]
        if len(mlist) == 0: return c
        p = max(mlist)
        for a in range(p,c,p):
            for m in mlist:
                if a % m and (c-a) % m:
                    break
            else:
                return c
        c += 1 # Chai Wah Wu, Oct 13 2021

Formula

a(n) <= A003418(n) and a(n) <= a(n+1). - David A. Corneth, Aug 11 2021

a(n) >= (4*A003418(n))^(1/3). - Charles R Greathouse IV, Oct 14 2021

Extensions

a(16)-a(26) from Alois P. Heinz, Aug 09 2021

a(27)-a(36) from David A. Corneth, Aug 11 2021

a(37)-a(40) from Chai Wah Wu, Oct 13 2021

a(41)-a(42) from Chai Wah Wu, Oct 21 2021