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 A333057 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. 2
 2024, 5624, 13688, 15375, 21608, 50300, 62775, 69375, 70784, 108927, 110888, 116864, 118016, 130815, 149768, 152703, 164024, 213759, 221823, 224720, 238975, 242432, 255231, 257175, 283904, 297135, 324224, 341887, 346544, 365295, 366848, 366975, 379647, 455552 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. LINKS Amiram Eldar, Table of n, a(n) for n = 1..10000 EXAMPLE 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]). MATHEMATICA f[p_, e_] := (e+1)*(e+2)/2;  d3 = 1; d3[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[10^5], d3[#] == d3[#+1] && Sort[FactorInteger[#][[;; , 2]]] != Sort[FactorInteger[#+1][[;; , 2]]] &] CROSSREFS Cf. A007425, A052213, A124010, A333055. Sequence in context: A013687 A257766 A126821 * A156855 A306870 A126172 Adjacent sequences:  A333054 A333055 A333056 * A333058 A333059 A333060 KEYWORD nonn AUTHOR Amiram Eldar, Mar 06 2020 STATUS approved

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Last modified August 15 11:04 EDT 2022. Contains 356145 sequences. (Running on oeis4.)