A350934 - OEIS

%I #16 Feb 20 2022 22:44:27

%S 34,9,7,964324,19,3822025,41,15129,341,427166224,199,700569,1241,

%T 11923111249,919,12376324,6641,34539129,12221,363016809,3401,56776225,

%U 5741,199809,52865,48045571249,47081,3764067904,21113,19035769,18089,145371249,59291,2219069449

%N a(n) is the smallest number m such that tau(m - 1) = tau(m + 1) = tau(m) + n or 0 if no such m exists, where tau(k) = A000005(k).

%C Corresponding values of tau(a(n)): 4, 3, 2, 9, 2, 27, 2, 9, 4, 15, 2, 21, 4, 27, 2, 9, 4, 15, 6, 45, 4, 27, 2, 9, 8, 15, 6, 21, 4, 3, 2, 9, 4, 15, ...

%C Triples of [tau(a(n) - 1), tau(a(n)), tau(a(n) + 1)] = [tau(a(n)) + n, tau(a(n)), tau(a(n)) + n]: [4, 4, 4], [4, 3, 4], [4, 2, 4], [12, 9, 12], [6, 2, 6], [32, 27, 32], [8, 2, 8], [16, 9, 16], [12, 4, 12], ...

%C If n is odd then a(n) is a square. - _Amiram Eldar_, Jan 26 2022

%e a(3) = 964324 because 964324 is the smallest number m such that tau(m-1) = tau(m+1) = tau(m)+3; tau(964323) = tau(964325) = tau(964324)+3 = 9+3 = 12.

%t seq[m_, nmax_] := Module[{s = Table[0, {m + 1}], c = 0, d1 = 1, d2 = 2, n = 3, d, k}, While[c < m + 1 && n < nmax, d = DivisorSigma[0, n]; If[d1 == d, k = d - d2 + 1; If[k >= 1 && k <= m + 1 && s[[k]] == 0, s[[k]] = n - 1; c++]]; n++; d1 = d2; d2 = d]; TakeWhile[s, # > 0 &]]; seq[8, 10^7] (* _Amiram Eldar_, Jan 26 2022 *)

%o (Magma) Ax:=func<n|exists(r){m: m in[2..10^6] | #Divisors(m - 1) eq #Divisors(m + 1) and #Divisors(m - 1) eq #Divisors(m) + n} select r else 0>; [Ax(n): n in [0..8]]

%Y Cf. A000005, A190646, A350132, A350935, A350936.

%K nonn

%O 0,1

%A _Jaroslav Krizek_, Jan 25 2022

%E More terms from _Amiram Eldar_, Jan 26 2022