A244053 Let m = A244052(n) = n-th highly regular number; a(n) = number of numbers r <= m, all of whose prime divisors p also divide m.

1, 2, 3, 5, 6, 8, 10, 11, 18, 19, 26, 28, 32, 36, 41, 44, 68, 77, 80, 96, 115, 131, 145, 156, 166, 174, 183, 192, 283, 295, 313, 322, 382, 395, 452, 463, 505, 519, 551, 567, 593, 629, 660, 691, 717, 743, 766, 1161, 1224, 1253, 1257, 1285, 1306, 1526

Offset

1,2

Comments

Analogous to A002183.

Records transform of A010846. -Michael De Vlieger, Mar 08 2017

Links

Michael De Vlieger and David A. Corneth, Table of n, a(n) for n = 1..563 (terms 55-149 from David A. Corneth)

Michael De Vlieger, Multiplicities of primes in the prime decomposition of A244052 and values of A244053.

Example

a(5) = 6 since 6 is the fifth record value of A010846. The first record value is 1 set at position 1; the second is 2 set at position 2, the third is 3 set at position 4, the fourth is 5 set at position 6. Sequence A244052 records the positions of these record values.

Mathematica

a010846[n_] := Block[{pf, a}, a[x_] := First /@ FactorInteger@ x; pf = a@ n; If[n == 1, 1, 1 + Count[Range@ n, _?(SubsetQ[pf, a@ #] &)]]]; f[n_] := Block[{t = {}, max = 0, x}, Do[If[(x = a010846@ i) > max, max = x; AppendTo[t, a010846[i]]], {i, n}]; t]; f@ 1000 (* Michael De Vlieger, Feb 10 2015 *)
Union@ Rest@ FoldList[Max, 0, Array[Count[Range@ #, k_ /; PowerMod[#, Floor@ Log2@ #, k] == 0] &, 10^3]] (* simplest, or *)
f[n_] := If[n == 1, 1, Length@ Function[w, ToExpression@ StringJoin["Module[{n = ", ToString@ n, ", k = 0}, Flatten@ Table[k++, ", Most@ Flatten@ Map[{#, ", "} &, #], "]]"] &@ MapIndexed[Function[p, StringJoin["{", ToString@ Last@ p, ", 0, Log[", ToString@ First@ p, ", n/(", ToString@ InputForm[Times @@ Map[Power @@ # &, Take[w, First@ #2 - 1]]], ")]}"]]@ w[[First@ #2]] &, w]]@ Map[{#, ToExpression["p" <> ToString@ PrimePi@ #]} &, FactorInteger[n][[All, 1]]]]; Union@ Rest@ FoldList[Max, 0, Array[f, 10^4]] (* Michael De Vlieger, Mar 08 2017, more efficient *)

Crossrefs

Cf. A244052, A002183, A010846.

Sequence in context: A005243 A117045 A324793 * A275833 A307113 A341157

Adjacent sequences: A244050 A244051 A244052 * A244054 A244055 A244056

Keyword

nonn

Author

Michael De Vlieger, Jun 18 2014

Status

approved