A343018 - OEIS

%I #22 Jul 26 2022 01:34:58

%S 2,1,5,49,11,35,23,399,47,1849,59,143,119,1599,167,575,179,1295,239,

%T 4355,629,2303,359,899,959,9215,1007,39999,719,20735,839,5183,1799,

%U 46655,1259,36863,1679,7055,3023,986049,2879,3599,6479,82943,2519,193599,3359,207935

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

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

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

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

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

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

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

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

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

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

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

%t 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 *)

%o (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]]

%o (PARI) a(n) = my(m=1); while (numdiv(m+1) != numdiv(m) + n, m++); m; \\ _Michel Marcus_, Apr 03 2021

%o (Python)

%o from itertools import count, pairwise

%o from sympy import divisor_count

%o 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

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

%K nonn

%O 0,1

%A _Jaroslav Krizek_, Apr 02 2021