A333056 Numbers k such that k, k+1 and k+2 have different prime signatures and d(k) = d(k+1) = d(k+2), where d(k) is the number of divisors of k (A000005).

59318, 72063, 72224, 184190, 185192, 215648, 300320, 355454, 362624, 384128, 548936, 550016, 640790, 682624, 707966, 723896, 758888, 828872, 828873, 858494, 860030, 888704, 901503, 963486, 963710, 993375, 1039742, 1039743, 1081214, 1248776, 1261897, 1340630

Offset: 1

Author

Amiram Eldar, Mar 06 2020

Keywords

• nonn

Comments

Apparently most of the numbers k such that d(k) = d(k+1) = d(k+2) (A005238) are terms of A052214, i.e., k, k+1 and k+2 have the same prime signature.

Of the first 10000 terms of A005238, 6406 are also in A052214, 3578 have a pair (k and k+1, k and k+2, or k+1 and k+2) with the same prime signature, and only 16 are in this sequence.

Examples

			59318 is a term since d(59318) = d(59319) = d(59320) = 16, and the prime signatures of these 3 numbers are different: 59318 = 2 * 7 * 19 * 223, 59319 = 3^3 * 13^3, and 59320 = 2^3 * 5 * 1483 have 3 different ordered prime signatures (A124010): [1, 1, 1, 1], [3, 3], and [1, 1, 3].
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A005238.

Cf. A000005, A052214, A124010, A333055.

Programs

Mathematica

psig[n_] := Sort @ FactorInteger[n][[;; , 2]]; d[sig_] := Times @@ (sig + 1); vsig = psig /@ Range[2, 4]; seqQ[v_] := Length@Union[v] == 3 && Length @ Union[d /@ v] == 1; seq = {}; Do[If[seqQ[vsig], AppendTo[seq, n - 3]]; vsig = Join[Rest[vsig], {psig[n]}], {n, 5, 10^6}]; seq