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%I #7 Mar 07 2020 03:51:42
%S 2024,5624,13688,15375,21608,50300,62775,69375,70784,108927,110888,
%T 116864,118016,130815,149768,152703,164024,213759,221823,224720,
%U 238975,242432,255231,257175,283904,297135,324224,341887,346544,365295,366848,366975,379647,455552
%N Numbers k such that k and k+1 have different (ordered) prime signatures and d_3(k) = d_3(k+1), where d_3 is A007425.
%C Apparently most of the numbers k such that k and k+1 have the same value of d_3 also have the same prime signature. a(1) = 2024 is the 212th number k such that d_3(k) = d_3(k+1), and up to 10^8 there are 8026247 such numbers k of them only 6414 are not in A052213.
%H Amiram Eldar, <a href="/A333057/b333057.txt">Table of n, a(n) for n = 1..10000</a>
%e 2024 is a term since d_3(2024) = d_3(2025) = 90, and the prime signatures of 2024 = 2^3 * 11 * 23 and 2025 = 3^4 * 5^2 are different ([1, 1, 3] and [2, 4]).
%t f[p_, e_] := (e+1)*(e+2)/2; d3[1] = 1; d3[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[10^5], d3[#] == d3[#+1] && Sort[FactorInteger[#][[;;,2]]] != Sort[FactorInteger[#+1][[;;,2]]] &]
%Y Cf. A007425, A052213, A124010, A333055.
%K nonn
%O 1,1
%A _Amiram Eldar_, Mar 06 2020