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A343018
a(n) is the smallest number m such that tau(m+1) = tau(m) + n.
3
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
0,1
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).
FORMULA
a(n) = A086550(n) - 1.
EXAMPLE
For n = 3; a(3) = 49 because 49 is the smallest number such that tau(50) = 6 = tau(49) + 3 = 3 + 3.
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 *)
PROG
(Magma) Ax:=func<n|exists(r){m: m in[1..10^6] | #Divisors(m + 1) - #Divisors(m) eq n} select r else 0>; [Ax(n): n in [0..50]]
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Jaroslav Krizek, Apr 02 2021
STATUS
approved