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

2,1

Comments

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

Links

Table of n, a(n) for n=2..42.

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

Example

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}.

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 *)

Prog

(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

Crossrefs

Cf. A003418, A027750, A346970.

Sequence in context: A296161 A121612 A061193 * A180941 A190896 A365018

Adjacent sequences: A346968 A346969 A346970 * A346972 A346973 A346974

Keyword

nonn,more

Author

Steven M. Altschuld, Aug 09 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

Status

approved