A083210 Numbers with no subset of their divisors such that the complement has the same sum.

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 81, 82, 83, 85, 86, 87, 89, 91

Offset: 1

Author

Reinhard Zumkeller, Apr 22 2003

Keywords

• nonn

Comments

A083206(a(n)) = 0; complement of A083207; deficient numbers (A005100) are a subset.

A179529(a(n)) = 0. [Reinhard Zumkeller, Jul 19 2010]

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Reinhard Zumkeller, Illustration of initial terms

Crossrefs

Positions of 0's in A083206.

Cf. A083207 (complement).

Cf. A005100, A083211 (subsequences).

Programs

PARI

is_A083210(n) = !A083206(n); \\ Antti Karttunen, Dec 02 2024

Python

from itertools import count, islice
from sympy import divisors
def A083210_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        d = divisors(n)
        s = sum(d)
        if s&1^1 and n<<1<=s:
            d = d[:-1]
            s2, ld = (s>>1)-n, len(d)
            z = [[0 for  in range(s2+1)] for  in range(ld+1)]
            for i in range(1, ld+1):
                y = min(d[i-1], s2+1)
                z[i][:y] = z[i-1][:y]
                for j in range(y,s2+1):
                    z[i][j] = max(z[i-1][j],z[i-1][j-y]+y)
                if z[i][s2] == s2:
                    break
            else:
                yield n
        else:
            yield n
A083210_list = list(islice(A083210_gen(),30)) # Chai Wah Wu, Feb 13 2023