A343019 a(n) is the smallest number m such that tau(m+1) = tau(m) - n.

2, 4, 6, 16, 12, 24, 30, 36, 84, 324, 60, 144, 192, 120, 210, 288, 180, 528, 240, 576, 480, 360, 420, 900, 1344, 960, 720, 5184, 1008, 840, 1320, 2400, 1260, 17424, 1800, 14640, 2640, 1680, 2160, 8280, 4800, 3600, 11220, 7056, 3780, 6240, 2520, 82944, 6480

Offset: 0

Author

Jaroslav Krizek, Apr 02 2021

Keywords

• nonn

Comments

tau(m) = the number of divisors of m (A000005).

A greedy inverse of A051950.

Sequences of numbers m such that tau(m+1) = tau(m) - n for 0 <= n <= 5:

n = 0: 2, 14, 21, 26, 33, 34, 38, 44, 57, 75, 85, 86, 93, ... (A005237).

n = 1: 4, 8, 81, 441, 625, 1089, 2024, 2401, 3025, 3968, ... (A068208).

n = 2: 6, 10, 20, 22, 32, 45, 46, 50, 58, 68, 76, 82, 92, ... (A227874).

n = 3: 16, 64, 224, 675, 1444, 2115, 3843, 5475, 6724, 9801, ...

n = 4: 12, 18, 28, 52, 54, 56, 105, 110, 114, 128, 148, 154, ...

n = 5: 24, 80, 225, 484, 1024, 1088, 1156, 1225, 1521, 2116, ...

Examples

			For n = 3; a(3) = 16 because 16 is the smallest number such that tau(17) = 2 = tau(16) - 3 = 5 - 3.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..174

Crossrefs

Cf. A000005 (tau), A051950, A080371, A080372, A343018.

Programs

Magma

Ax:=func; [Ax(n): n in [0..50]]

Maple

N:= 50: # for a(0)..a(N)
V:= Array(0..N): count:=0: t:= numtheory:-tau(1):
for m from 1 while count < N+1 do
  s:= numtheory:-tau(m+1); v:= t - s;
  if v >= 0 and v <= N and V[v] = 0 then
    count:= count+1; V[v]:= m;
  fi;
  t:= s;
od:
convert(V,list); # Robert Israel, Jul 03 2024

Mathematica

d = Differences @ Table[DivisorSigma[0, n], {n, 1, 10^5}]; a[n_] := If[(p = Position[d, -n]) != {}, p[[1, 1]], 0]; s = {}; n = 0; While[(a1 = a[n]) > 0, AppendTo[s, a1]; n++]; s (* Amiram Eldar, Apr 03 2021 *)

PARI

a(n) = my(m=1); while (numdiv(m+1) != numdiv(m) - n, m++); m; \\ Michel Marcus, Apr 03 2021

Python

from itertools import count, pairwise
from sympy import divisor_count
def A343019(n): return next(m+1 for m, t in enumerate(pairwise(map(divisor_count,count(1)))) if t[1] == t[0]-n) # Chai Wah Wu, Jul 25 2022