A383179 Numbers k such that omega(k) = 5 and p^omega(k) < k^(1/5) < lpf(k)^(omega(k)+1) for all primes p | k such that p > lpf(k), where lpf = A020639(k).

101007559, 112442377, 145352341, 370621421, 392748073, 396181519, 403811399, 496492847, 510478561, 530733733, 540954893, 545683979, 552435703, 578262127, 580407131, 585416939, 590534717, 594163571, 620435209, 625790521, 633456391, 635140369, 643418423, 651300233

Offset: 1

Author

Michael De Vlieger, May 09 2025

Keywords

• nonn

Comments

A010846(a(n)) >= 176.

Examples

			Table of n, a(n), prime decomposition of a(n), and A010846(n) = c(n) for n = 1..12 and n = 209 (the smallest term with c(n) = 176):
  n         a(n)    facs(a(n))    c(a(n))
--------------------------------------
  1   101007559   23*41*43*47*53    180
  2   112442377   23*41*43*47*59    182
  3   145352341   23*43*47*53*59    179
  4   370621421   29*53*59*61*67    179
  5   392748073   29*53*59*61*71    180
  6   396181519   31*53*59*61*67    179
  7   403811399   29*53*59*61*73    181
  8   496492847   29*59*61*67*71    179
  9   510478561   29*59*61*67*73    179
 10   530733733   31*59*61*67*71    179
 11   540954893   29*59*61*71*73    179
 12   545683979   31*59*61*67*73    179
209  3433936673   41*83*97*101*103  176
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Plot prime(i) | a(n) at (x,y) = (n,i) for n = 1..2048, 8X vertical exaggeration. The green bar at the bottom of the graph emphasizes the x axis that rides on the top edge of the bar.

Crossrefs

Cf. A010846, A020639, A162306, A383177, A383178.

Programs

Mathematica

f[om_, lm_ : 0] := Block[{f, i, j, k, nn, w}, i = Abs[om]; j = 1;
  If[lm == 0, nn = Times @@ Prime@ Range[i], nn = Abs[lm]]; w = ConstantArray[1, i];
  Union@ Reap[Do[
    While[Set[k, Times @@ Map[Prime, Accumulate@w]]; k <= nn,
      If[Or[k == 1, Union[#2] == #1 - 1 & @@
        TakeDrop[Map[Floor@Log[#, k] &, FactorInteger[k][[All, 1]] ], 1] ],
        Sow[k]];
      j = 1; w[[-j]]++];
      If[j == i, Break[], j++; w[[-j]]++;
        w = PadRight[w[[;; -j]], i, 1]], {n, Infinity}] ][[-1, 1]] ];
f[5, 10^9, 5]