A343018 a(n) is the smallest number m such that tau(m+1) = tau(m) + n.

2, 1, 5, 49, 11, 35, 23, 399, 47, 1849, 59, 143, 119, 1599, 167, 575, 179, 1295, 239, 4355, 629, 2303, 359, 899, 959, 9215, 1007, 39999, 719, 20735, 839, 5183, 1799, 46655, 1259, 36863, 1679, 7055, 3023, 986049, 2879, 3599, 6479, 82943, 2519, 193599, 3359, 207935

Offset: 0

Author

Jaroslav Krizek, Apr 02 2021

Keywords

• nonn

Comments

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

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: 1, 3, 9, 15, 25, 63, 121, 195, 255, 361, 483, 729, ... (A055927).

n = 2: 5, 7, 13, 27, 37, 51, 61, 62, 73, 74, 91, 115, 123, ... (A230115).

n = 3: 49, 99, 1023, 1681, 1935, 2499, 8649, 9603, 20449, ... (A230653).

n = 4: 11, 17, 19, 31, 39, 43, 55, 65, 67, 69, 77, 87, 97, ... (A230654).

n = 5: 35, 169, 289, 529, 961, 1369, 2809, 3135, 4489, ... (A228453).

Examples

			For n = 3; a(3) = 49 because 49 is the smallest number such that tau(50) = 6 = tau(49) + 3 = 3 + 3.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..188

Crossrefs

Cf. A000005 (tau), A080371, A080372, A086550, A340799, A343019, A343020.

Programs

Magma

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

Maple

N:= 60: # 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:= s - t;
  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, Jan 03 2025

Mathematica

d = Differences @ Table[DivisorSigma[0, n], {n, 1, 10^6}]; 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 A343018(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

Formula

a(n) = A086550(n) - 1.