A350934 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).

34, 9, 7, 964324, 19, 3822025, 41, 15129, 341, 427166224, 199, 700569, 1241, 11923111249, 919, 12376324, 6641, 34539129, 12221, 363016809, 3401, 56776225, 5741, 199809, 52865, 48045571249, 47081, 3764067904, 21113, 19035769, 18089, 145371249, 59291, 2219069449

Offset

0,1

Comments

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, ...

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], ...

If n is odd then a(n) is a square. - Amiram Eldar, Jan 26 2022

Links

Table of n, a(n) for n=0..33.

Example

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.

Mathematica

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 *)

Prog

(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]]

Crossrefs

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

Sequence in context: A298138 A350935 A350132 * A071788 A220133 A300101

Adjacent sequences: A350931 A350932 A350933 * A350935 A350936 A350937

Keyword

nonn

Author

Jaroslav Krizek, Jan 25 2022

Extensions

More terms from Amiram Eldar, Jan 26 2022

Status

approved