A356660 Numbers k that can be written as the sum of 10 divisors of k (not necessarily distinct).

10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 40, 42, 44, 48, 50, 52, 54, 56, 60, 64, 66, 68, 70, 72, 76, 78, 80, 84, 88, 90, 92, 96, 98, 100, 102, 104, 108, 110, 112, 114, 116, 120, 126, 128, 130, 132, 136, 138, 140, 144, 150, 152, 154, 156, 160, 162

Offset

1,1

Comments

From David A. Corneth, Oct 08 2022: (Start)

All terms are even. Proof: suppose a term is odd. Then all divisors are odd. Adding 10 odd numbers gives an even number. A contradiction.

If k is a term then so is k*m for m >= 1. Proof: Multiply each divisor in this sum of 10 divisors that give k with m. Then each term is a divisor of k*m and their sum is k*m. (End)

Links

David A. Corneth, Table of n, a(n) for n = 1..10000

Example

14 is in the sequence since 14 = 2+2+2+2+1+1+1+1+1+1, where each summand divides 14.

Prog

(Python)
from sympy import divisors
def t_sum_of_n_div(n, target):
    out, p = [], divisors(n)[::-1][1:]
    def dfs(t, divs,  index_s, kk):
        if len(out)!=0 or kk>target:return
        if kk == target and t == 0:
            out.append(divs)
            return
        for i in range(index_s, len(p)):
            if t >= p[i]:
                temp_divs = divs.copy()
                temp_divs.append(p[i])
                dfs(t-p[i], temp_divs, i, kk+1)
    dfs(n, [], 0, 0)
    return out
terms = [i for i in range(2, 200) if len(t_sum_of_n_div(i, 10))!=0]
print(terms) # Gleb Ivanov, Sep 02 2022
(PARI) upto(n) = { my(v = vector(n, i, -1), t = 0); forstep(i = 2, n, 2, if(v[i] == -1, v[i] = is(i); if(v[i] == 1, for(j = 2, n \ i, v[i*j] = 1; ) ) ); ); select(x->x >= 1, v, 1); }
is(n, {qd = 10}) = { my(d = divisors(n), res = 0); d = d[^#d]; forvec(x = vector(qd-1, i, [1, #d]), s = sum(i = 1, qd-1, d[x[i]]); if(n - s >= d[x[qd - 1]], if(n % (n - s) == 0,  return(1); ) ) , 1 ); 0 } \\ David A. Corneth, Oct 08 2022

Crossrefs

Numbers k that can be written as the sum of j divisors of k (not necessarily distinct) for j=1..10: A000027 (j=1), A299174 (j=2), A355200 (j=3), A354591 (j=4), A355641 (j=5), A356609 (j=6), A356635 (j=7), A356657 (j=8), A356659 (j=9), this sequence (j=10).

Sequence in context: A351998 A088381 A163750 * A167153 A298298 A247393

Adjacent sequences: A356657 A356658 A356659 * A356661 A356662 A356663

Keyword

nonn

Author

Wesley Ivan Hurt, Aug 20 2022

Status

approved