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%I #11 Mar 07 2020 03:51:45
%S 59318,72063,72224,184190,185192,215648,300320,355454,362624,384128,
%T 548936,550016,640790,682624,707966,723896,758888,828872,828873,
%U 858494,860030,888704,901503,963486,963710,993375,1039742,1039743,1081214,1248776,1261897,1340630
%N 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).
%C 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.
%C 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.
%H Amiram Eldar, <a href="/A333056/b333056.txt">Table of n, a(n) for n = 1..10000</a>
%e 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].
%t 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
%Y Subsequence of A005238.
%Y Cf. A000005, A052214, A124010, A333055.
%K nonn
%O 1,1
%A _Amiram Eldar_, Mar 06 2020
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