A333055 -id:A333055 - cp's OEIS frontend

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

Amiram Eldar, Mar 06 2020

nonn

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.

			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].

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

Subsequence of A005238.

Cf. A000005, A052214, A124010, A333055.

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

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.

Original entry on oeis.org

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

Amiram Eldar, Mar 06 2020

nonn

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.

			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]).

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

Cf. A007425, A052213, A124010, A333055.

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]]] &]

cp's OEIS Frontend

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).

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