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
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}.
Programs
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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
Comments