A375574 Let d(1)<d(2)<...<d(q) denote the divisors of an integer k. a(n) = k is the smallest k such that the sum of its first n divisors, s = d(1) + ... + d(n), is also a divisor of k.

1, 6, 6, 28, 28, 24, 126, 234, 224, 360, 504, 980, 990, 1260, 1764, 1680, 840, 1080, 4140, 960, 5760, 4620, 9180, 11088, 8960, 6120, 11880, 25740, 7140, 2520, 2016, 25344, 9720, 48672, 11760, 10920, 15120, 14112, 61740, 55200, 74340, 91800, 8190, 78624, 70200

Offset

1,2

Comments

The index i of s among the divisors of k is i = A375593(n), i.e. s = d(A375593(n)).

Links

Table of n, a(n) for n=1..45.

David A. Corneth, PARI program

Example

*----*------*---------*---------------------------------*
| n  | a(n) |  i |  i-th   |  sum of n first divisors   |
|    |      |    | divisor |         of a(n)            |
*----*------*---------*---------------------------------*
| 2  |   6  |  3 |    3    | 1+2 = 3                    |
*----*------*----*---------*----------------------------*
| 3  |   6  |  4 |    6    | 1+2+3 = 6                  |
*----*------*----*---------*----------------------------*
| 4  |  28  |  5 |   14    | 1+2+4+7 = 14               |
*----*------*----*---------*----------------------------*
| 5  |  28  |  6 |   28    | 1+2+4+7+14 = 28            |
*----*------*----*---------*----------------------------*
| 6  |  24  |  8 |   24    | 1+2+3+4+6+8 = 24           |
*----*------*----*---------*----------------------------*
| 7  | 126  | 10 |   42    | 1+2+3+6+7+9+14 = 42        |
*----*------*----*---------*----------------------------*
| 8  | 234  | 10 |   78    | 1+2+3+6+9+13+18+26 = 78    |
*----*------*----*---------*----------------------------*
| 9  | 224  | 11 |  112    | 1+2+4+7+8+14+16+28+32 = 112|
|----*------*----*---------*----------------------------*

Maple

with(numtheory):nn:=10^6:T:=array(1..44):i:=0:
for n from 2 to 45 do:
 ii:=1:
  for a from 6 to nn while ii=1
do:
    d:=divisors(a):n0:=nops(d):
     if n0>=n
      then
       s:=sum('d[j]', 'j'=1..n):
       for m from 1 to n0 do:
        if s=d[m]
         then
          ii:=0:printf(`%d %d\n`, n, a):i:=i+1:T[i]:=a:
           else
          fi :
        od :fi:
  od:od:print(T):

Prog

(PARI) \\ See Corneth link
(Python)
from sympy import divisors
from itertools import count, islice
def agen(): # generator of terms
    adict, n = dict(), 1
    for k in count(1):
        d = divisors(k)
        if len(d) < n-1: continue
        dset, s = set(d), 0
        for i, di in enumerate(d, 1):
            s += di
            if i >= n and i not in adict and s in dset: adict[i] = k
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 50))) # Michael S. Branicky, Aug 20 2024

Crossrefs

Cf. A000005, A000203, A001065, A240698, A375593.

Sequence in context: A274940 A341548 A253066 * A267651 A016725 A151779

Adjacent sequences: A375571 A375572 A375573 * A375575 A375576 A375577

Keyword

nonn

Author

Michel Lagneau, Aug 19 2024

Extensions

a(1) = 1 prepended by David A. Corneth, Aug 20 2024

Status

approved