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.

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

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] = 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: A257766 A359839 A126821 * A156855 A306870 A126172

Adjacent sequences: A333054 A333055 A333056 * A333058 A333059 A333060

Keyword

nonn

Author

Amiram Eldar, Mar 06 2020

Status

approved