A381240 Indices of records in A381096.

1, 6, 10, 12, 14, 18, 22, 24, 28, 30, 42, 54, 60, 66, 78, 84, 90, 102, 114, 120, 126, 132, 138, 150, 168, 180, 198, 204, 210, 240, 252, 264, 270, 294, 300, 330, 360, 378, 390, 420, 450, 462, 480, 504, 510, 540, 546, 570, 600, 630, 660, 690, 714, 750, 780, 810, 840

Offset: 1

Author

Michael De Vlieger, Feb 18 2025

Keywords

• nonn

Comments

Let f(k) = A381096(k) = k - phi(k) - tau(k/rad(k)) = k - A000010(k) - A005361(k), where phi = A000010, tau = A000005, and rad = A007947. This sequence contains k such that f(k) > f(j) for j < k as k increases.

Apart from a(1) = 1, terms are in A024619.

Conjecture 1: For i > 1, A002110(i) is in this sequence.

Conjecture 2: Intersection with A001694 (i.e., in A286708) is {900, 1800}.

Examples

			Let g(n) = A067255(n) be the exponents of prime power factors p^m | n, writing "." for m = 0 and ending at the pi(gpf(n))-th term. Example: for n = 84, g(84) = {2, 1, 0, 1}, therefore we write "21.1" for concision in the table below.
Table of first 12 terms.
   n  a(n)  g(a(n)) f(a(n))
  --------------------------
   1    1   .           0
   2    6   11          3
   3   10   1.1         5
   4   12   21          6
   5   14   1..1        7
   6   18   12         10
   7   22   1...1      11
   8   24   31         13
   9   28   2..1       14
  10   30   111        21
  11   42   11.1       29
  12   54   13         33
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..1782 (a(n) < 2^28).
• Michael De Vlieger, Plot p^m | a(n) at (x,y) = (n,pi(p)), n = 1..1781, 4X vertical exaggeration for clarity, with a color function representing m = 1 in black, m = 2 in red, m = 3 in orange, ..., largest m in the dataset in magenta.

Crossrefs

Cf. A000010, A005361, A024619, A381096, A381241.

Programs

Mathematica

r = 0; nn = 2^20; f[x_] := x - EulerPhi[x] - DivisorSigma[0, x/Apply[Times, FactorInteger[x][[All, 1]] ] ]; {1}~Join~Reap[Monitor[Do[If[# > r, r = #; Sow[n]] &[f[n] ], {n, nn}], n]]