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A375574 Let d(1) 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.

%I A375574 #21 Sep 13 2024 07:43:42
%S A375574 1,6,6,28,28,24,126,234,224,360,504,980,990,1260,1764,1680,840,1080,
%T A375574 4140,960,5760,4620,9180,11088,8960,6120,11880,25740,7140,2520,2016,
%U A375574 25344,9720,48672,11760,10920,15120,14112,61740,55200,74340,91800,8190,78624,70200
%N 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.
%C A375574 The index i of s among the divisors of k is i = A375593(n), i.e. s = d(A375593(n)).
%H A375574 David A. Corneth, <a href="/A375574/a375574.gp.txt">PARI program</a>
%e A375574 *----*------*---------*---------------------------------*
%e A375574 | n  | a(n) |  i |  i-th   |  sum of n first divisors   |
%e A375574 |    |      |    | divisor |         of a(n)            |
%e A375574 *----*------*---------*---------------------------------*
%e A375574 | 2  |   6  |  3 |    3    | 1+2 = 3                    |
%e A375574 *----*------*----*---------*----------------------------*
%e A375574 | 3  |   6  |  4 |    6    | 1+2+3 = 6                  |
%e A375574 *----*------*----*---------*----------------------------*
%e A375574 | 4  |  28  |  5 |   14    | 1+2+4+7 = 14               |
%e A375574 *----*------*----*---------*----------------------------*
%e A375574 | 5  |  28  |  6 |   28    | 1+2+4+7+14 = 28            |
%e A375574 *----*------*----*---------*----------------------------*
%e A375574 | 6  |  24  |  8 |   24    | 1+2+3+4+6+8 = 24           |
%e A375574 *----*------*----*---------*----------------------------*
%e A375574 | 7  | 126  | 10 |   42    | 1+2+3+6+7+9+14 = 42        |
%e A375574 *----*------*----*---------*----------------------------*
%e A375574 | 8  | 234  | 10 |   78    | 1+2+3+6+9+13+18+26 = 78    |
%e A375574 *----*------*----*---------*----------------------------*
%e A375574 | 9  | 224  | 11 |  112    | 1+2+4+7+8+14+16+28+32 = 112|
%e A375574 |----*------*----*---------*----------------------------*
%p A375574 with(numtheory):nn:=10^6:T:=array(1..44):i:=0:
%p A375574 for n from 2 to 45 do:
%p A375574  ii:=1:
%p A375574   for a from 6 to nn while ii=1
%p A375574 do:
%p A375574     d:=divisors(a):n0:=nops(d):
%p A375574      if n0>=n
%p A375574       then
%p A375574        s:=sum('d[j]', 'j'=1..n):
%p A375574        for m from 1 to n0 do:
%p A375574         if s=d[m]
%p A375574          then
%p A375574           ii:=0:printf(`%d %d\n`,n,a):i:=i+1:T[i]:=a:
%p A375574            else
%p A375574           fi :
%p A375574         od :fi:
%p A375574   od:od:print(T):
%o A375574 (PARI) \\ See Corneth link
%o A375574 (Python)
%o A375574 from sympy import divisors
%o A375574 from itertools import count, islice
%o A375574 def agen(): # generator of terms
%o A375574     adict, n = dict(), 1
%o A375574     for k in count(1):
%o A375574         d = divisors(k)
%o A375574         if len(d) < n-1: continue
%o A375574         dset, s = set(d), 0
%o A375574         for i, di in enumerate(d, 1):
%o A375574             s += di
%o A375574             if i >= n and i not in adict and s in dset: adict[i] = k
%o A375574         while n in adict: yield adict[n]; n += 1
%o A375574 print(list(islice(agen(), 50))) # _Michael S. Branicky_, Aug 20 2024
%Y A375574 Cf. A000005, A000203, A001065, A240698, A375593.
%K A375574 nonn
%O A375574 1,2
%A A375574 _Michel Lagneau_, Aug 19 2024
%E A375574 a(1) = 1 prepended by _David A. Corneth_, Aug 20 2024