A081513
a(2)=0; for n != 2, let m = A081512(n), then a(n) = value of d_n in the lexicographically earliest set of n distinct divisors d_1, ..., d_n of m with d_1 + ... + d_n = m.
Original entry on oeis.org
1, 0, 3, 6, 12, 8, 24, 20, 28, 40, 40, 30, 60, 45, 60, 120, 90, 72, 72, 168, 240, 180, 144, 168, 168, 216, 420, 315, 210, 420, 420, 336, 432, 1260, 840, 840, 504, 504, 315, 480, 540, 2520, 2520, 1680, 1680, 1260, 1260, 1008, 720, 2520, 2520, 1890, 1512, 945, 3360
Offset: 1
The lexicographically earliest solutions are:
..n....m: d_1 d_2 ... d_n
-------------------------
..1....1: 1
..2....0: - -
..3....6: 1, 2, 3
..4...12: 1, 2, 3, 6
..5...24: 1, 2, 3, 6, 12
..6...24: 1, 2, 3, 4, 6, 8
..7...48: 1, 2, 3, 4, 6, 8, 24
..8...60: 1, 2, 3, 4, 5, 10, 15, 20
..9...84: 1, 2, 3, 4, 6, 7, 12, 21, 28
.10..120: 1, 2, 3, 4, 5, 6, 15, 20, 24, 40
...
Corrected by Caleb M. Shor (cshor(AT)bates.edu), Sep 26 2007
A081514
Triangle read by rows: row n = lexicographically earliest choice for n distinct divisors of A081512(n) = m whose sum is m.
Original entry on oeis.org
1, 0, 0, 1, 2, 3, 1, 2, 3, 6, 1, 2, 3, 6, 12, 1, 2, 3, 4, 6, 8, 1, 2, 3, 4, 6, 8, 24, 1, 2, 3, 4, 5, 10, 15, 20, 1, 2, 3, 4, 6, 7, 12, 21, 28, 1, 2, 3, 4, 5, 6, 15, 20, 24, 40, 1, 2, 3, 4, 5, 6, 8, 12, 15, 24, 40, 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 24, 30, 1, 2, 3, 4, 5, 6, 9, 10, 12, 18, 20, 30, 60
Offset: 1
The lexicographically earliest solutions are:
[1]
[0, 0]
[1, 2, 3]
[1, 2, 3, 6]
[1, 2, 3, 6, 12]
[1, 2, 3, 4, 6, 8]
[1, 2, 3, 4, 6, 8, 24]
[1, 2, 3, 4, 5, 10, 15, 20]
[1, 2, 3, 4, 6, 7, 12, 21, 28]
[1, 2, 3, 4, 5, 6, 15, 20, 24, 40]
[1, 2, 3, 4, 5, 6, 8, 12, 15, 24, 40]
[1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 24, 30]
[1, 2, 3, 4, 5, 6, 9, 10, 12, 18, 20, 30, 60]
...
Corrected by Caleb M. Shor (cshor(AT)bates.edu), Sep 26 2007
Original entry on oeis.org
1, 0, 6, 12, 24, 24, 48, 60, 84, 120, 120, 120, 180, 180, 240, 360, 360, 360, 360, 672, 720, 720, 720, 840, 840, 1080, 1260, 1260, 1260, 1680, 1680, 1680, 2160, 2520, 2520, 2520, 2520, 2520, 2520, 3360, 4320, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040
Offset: 1
A377247
a(n) is the largest k such that the sum of the first k divisors of n is at most n.
Original entry on oeis.org
1, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 4, 1, 3, 3, 4, 1, 4, 1, 4, 3, 3, 1, 6, 2, 3, 3, 5, 1, 6, 1, 5, 3, 3, 3, 6, 1, 3, 3, 6, 1, 6, 1, 5, 5, 3, 1, 7, 2, 5, 3, 5, 1, 6, 3, 6, 3, 3, 1, 9, 1, 3, 5, 6, 3, 6, 1, 5, 3, 6, 1, 9, 1, 3, 5, 5, 3, 6, 1, 8, 4, 3, 1, 9, 3, 3, 3, 6
Offset: 1
a(1) = 1 as the sum of the first divisor of 1 is 1 <= 1 and 1 has no more divisors.
a(6) = 3 as the sum of the first three divisors is 1+2+3 <= 6 but the sum of the first four divisors is 1 + 2 + 3 + 6 = 12 > 6.
-
A377247[n_] := LengthWhile[Accumulate[Divisors[n]], # <= n &];
Array[A377247, 100] (* Paolo Xausa, Aug 05 2025 *)
-
A377247(n) = {my(d = divisors(n), t = 0); for(i = 1, #d, t += d[i]; if(t > n, return(i-1))); 1}
A355228
a(n) is the smallest integer m such that there exist n of its distinct divisors (d_1, d_2, ..., d_n) with the property that m = d_1 + d_2 + ... + d_n = lcm(d_1, d_2, ..., d_n), or 0 if no such number m exists.
Original entry on oeis.org
1, 0, 6, 18, 28, 24, 48, 60, 84, 120, 120, 120, 180, 180, 240, 360, 360, 360, 360, 672, 720, 720, 720, 840, 840, 1080, 1260, 1260, 1260, 1680, 1680, 1680, 2160, 2520, 2520, 2520, 2520, 2520, 2520, 3360, 4320, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040
Offset: 1
In the following triangle, the n-th row gives an example of a set of n divisors d_1, ..., d_n of a(n) such that a(n) = d_1 + ... + d_n = lcm(d_1, ..., d_n):
.
n m d_1 d_2 d_3 d_4 d_5 d_6 d_7 d_8 d_9 d10 d11 d12
-----------------------------------------------------------
1 1 1
2 0
3 6 1 2 3
4 18 1 2 6 9
5 28 1 2 4 7 14
6 24 1 2 3 4 6 8
7 48 1 2 3 4 8 16 24
8 60 1 2 3 4 5 10 15 20
9 84 1 2 3 4 6 7 12 21 28
10 120 1 2 3 4 5 6 15 20 24 40
11 120 1 2 3 4 5 6 8 12 15 24 40
12 120 1 2 3 4 5 6 8 10 12 15 24 30
However, for a given value of a(n) = m, there may be more than one way to choose d_1, ..., d_n. For example, for n=10, a(10)=120 and all seventeen solutions provided by _Jinyuan Wang_ in the Comments section of A081512 are also solutions here.
Showing 1-5 of 5 results.
Comments