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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).

%I A383179 #7 May 16 2025 00:55:47
%S A383179 101007559,112442377,145352341,370621421,392748073,396181519,
%T A383179 403811399,496492847,510478561,530733733,540954893,545683979,
%U A383179 552435703,578262127,580407131,585416939,590534717,594163571,620435209,625790521,633456391,635140369,643418423,651300233
%N 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).
%C A383179 A010846(a(n)) >= 176.
%H A383179 Michael De Vlieger, <a href="/A383179/b383179.txt">Table of n, a(n) for n = 1..10000</a>
%H A383179 Michael De Vlieger, <a href="/A383179/a383179.png">Plot prime(i) | a(n) at (x,y) = (n,i)</a> 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.
%e A383179 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):
%e A383179   n         a(n)    facs(a(n))    c(a(n))
%e A383179 --------------------------------------
%e A383179   1   101007559   23*41*43*47*53    180
%e A383179   2   112442377   23*41*43*47*59    182
%e A383179   3   145352341   23*43*47*53*59    179
%e A383179   4   370621421   29*53*59*61*67    179
%e A383179   5   392748073   29*53*59*61*71    180
%e A383179   6   396181519   31*53*59*61*67    179
%e A383179   7   403811399   29*53*59*61*73    181
%e A383179   8   496492847   29*59*61*67*71    179
%e A383179   9   510478561   29*59*61*67*73    179
%e A383179  10   530733733   31*59*61*67*71    179
%e A383179  11   540954893   29*59*61*71*73    179
%e A383179  12   545683979   31*59*61*67*73    179
%e A383179 209  3433936673   41*83*97*101*103  176
%t A383179 f[om_, lm_ : 0] := Block[{f, i, j, k, nn, w}, i = Abs[om]; j = 1;
%t A383179   If[lm == 0, nn = Times @@ Prime@ Range[i], nn = Abs[lm]]; w = ConstantArray[1, i];
%t A383179   Union@ Reap[Do[
%t A383179     While[Set[k, Times @@ Map[Prime, Accumulate@w]]; k <= nn,
%t A383179       If[Or[k == 1, Union[#2] == #1 - 1 & @@
%t A383179         TakeDrop[Map[Floor@Log[#, k] &, FactorInteger[k][[All, 1]] ], 1] ],
%t A383179         Sow[k]];
%t A383179       j = 1; w[[-j]]++];
%t A383179       If[j == i, Break[], j++; w[[-j]]++;
%t A383179         w = PadRight[w[[;; -j]], i, 1]], {n, Infinity}] ][[-1, 1]] ];
%t A383179 f[5, 10^9, 5]
%Y A383179 Cf. A010846, A020639, A162306, A383177, A383178.
%K A383179 nonn
%O A383179 1,1
%A A383179 _Michael De Vlieger_, May 09 2025