cp's OEIS Frontend

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

%I A343018 #30 Jan 03 2025 11:47:34
%S A343018 2,1,5,49,11,35,23,399,47,1849,59,143,119,1599,167,575,179,1295,239,
%T A343018 4355,629,2303,359,899,959,9215,1007,39999,719,20735,839,5183,1799,
%U A343018 46655,1259,36863,1679,7055,3023,986049,2879,3599,6479,82943,2519,193599,3359,207935
%N A343018 a(n) is the smallest number m such that tau(m+1) = tau(m) + n.
%C A343018 tau(m) = the number of divisors of m (A000005).
%C A343018 Sequences of numbers m such that tau(m+1) = tau(m) + n for 0 <= n <= 5:
%C A343018 n = 0: 2, 14, 21, 26, 33, 34, 38, 44, 57, 75, 85, 86, 93, ... (A005237).
%C A343018 n = 1: 1, 3, 9, 15, 25, 63, 121, 195, 255, 361, 483, 729, ... (A055927).
%C A343018 n = 2: 5, 7, 13, 27, 37, 51, 61, 62, 73, 74, 91, 115, 123, ... (A230115).
%C A343018 n = 3: 49, 99, 1023, 1681, 1935, 2499, 8649, 9603, 20449, ... (A230653).
%C A343018 n = 4: 11, 17, 19, 31, 39, 43, 55, 65, 67, 69, 77, 87, 97, ... (A230654).
%C A343018 n = 5: 35, 169, 289, 529, 961, 1369, 2809, 3135, 4489, ... (A228453).
%H A343018 Robert Israel, <a href="/A343018/b343018.txt">Table of n, a(n) for n = 0..188</a>
%F A343018 a(n) = A086550(n) - 1.
%e A343018 For n = 3; a(3) = 49 because 49 is the smallest number such that tau(50) = 6 = tau(49) + 3 = 3 + 3.
%p A343018 N:= 60: # for a(0)..a(N)
%p A343018 V:= Array(0..N): count:=0: t:= numtheory:-tau(1):
%p A343018 for m from 1 while count < N+1 do
%p A343018   s:= numtheory:-tau(m+1); v:= s - t;
%p A343018   if v >= 0 and v <= N and V[v] = 0 then count:= count+1; V[v]:= m; fi;
%p A343018   t:= s;
%p A343018 od:
%p A343018 convert(V, list); # _Robert Israel_, Jan 03 2025
%t A343018 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 A343018 (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 A343018 (PARI) a(n) = my(m=1); while (numdiv(m+1) != numdiv(m) + n, m++); m; \\ _Michel Marcus_, Apr 03 2021
%o A343018 (Python)
%o A343018 from itertools import count, pairwise
%o A343018 from sympy import divisor_count
%o A343018 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 A343018 Cf. A000005 (tau), A080371, A080372, A086550, A340799, A343019, A343020.
%K A343018 nonn
%O A343018 0,1
%A A343018 _Jaroslav Krizek_, Apr 02 2021