This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A383178 #7 May 16 2025 00:55:54
%S A383178 81719,268801,565471,626603,631997,657169,700321,799459,838457,893513,
%T A383178 916453,1108927,1212083,1239389,1271209,1354681,1366817,1408637,
%U A383178 1420763,1500313,1527619,1574359,1602137,1639877,1700557,1719871,1751173,1758203,1775341,1783511,1843969
%N A383178 Numbers k such that omega(k) = 4 and p^omega(k) < k^(1/4) < lpf(k)^(omega(k)+1) for all primes p | k such that p > lpf(k), where lpf = A020639(k).
%C A383178 Let primes p, q, r, s, p < q < r < s, divide k = a(n).
%C A383178 Then floor(log(k)/log(p)) = 4 and floor(log(k)/log(q)) = floor(log(k)/log(r)) = floor(log(k)/log(s)) = 3.
%C A383178 Let R(k) = row k of A162306 = {m <= k : rad(m) | k}. Then A010846(k) = c(k) is the number of terms in row k of A162306.
%C A383178 A010846(a(n)) = 51 for k such that p*r^3 < k.
%C A383178 A010846(a(n)) = 50 for k such that p*r^3 > k.
%H A383178 Michael De Vlieger, <a href="/A383178/b383178.txt">Table of n, a(n) for n = 1..10000</a>
%H A383178 Michael De Vlieger, <a href="/A383178/a383178.png">Hasse diagram of R(268801)</a> with logarithmic vertical scale. Gray represents the empty product, red represents primes, gold represents proper prime powers, green squarefree composites, and blue numbers that are neither squarefree nor prime powers.
%H A383178 Michael De Vlieger, <a href="/A383178/a383178_1.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 A383178 Table of n, a(n), prime decomposition of a(n), and A010846(n) = c(n) for n = 1..12:
%e A383178 n a(n) facs(a(n)) c(n) p*r^3
%e A383178 --------------------------------------------------
%e A383178 1 81719 11*17*19*23 51 11*19^3 = 75449
%e A383178 2 268801 13*23*29*31 50 13*29^3 = 317057
%e A383178 3 565471 17*29*31*37 51 17*31^3 = 506447
%e A383178 4 626603 17*29*31*41 51 17*31^3 = 506447
%e A383178 5 631997 19*29*31*37 51 19*31^3 = 566029
%e A383178 6 657169 17*29*31*43 51 17*31^3 = 506447
%e A383178 7 700321 19*29*31*41 51 19*31^3 = 566029
%e A383178 8 799459 17*31*37*41 50 17*37^3 = 861101
%e A383178 9 838457 17*31*37*43 50 17*37^3 = 861101
%e A383178 10 893513 19*31*37*41 50 19*37^3 = 962407
%e A383178 11 916453 17*31*37*47 51 17*37^3 = 861101
%e A383178 12 1108927 17*37*41*43 50 17*41^3 = 1171657
%e A383178 Let S(k) = row k of A162306 = {m <= k : rad(m) | k}.
%e A383178 Writing p^a*q^b*r^c*s^d instead as "abcd" (i.e., catenating prime power exponents), the following combinations are in S(a(n)). In brackets we show p*r^3, which is in S(a(n)) for n such that c(n) = 51, but not in S(a(n)) for n such that c(n) = 50.
%e A383178 0000 1000 2000 3000 4000 0010 1010 2010 3010 0020 1020 2020 0030 [1030]
%e A383178 0100 1100 2100 3100 0110 1110 2110 0120 1120
%e A383178 0200 1200 2200 0210 1210
%e A383178 0300 1300
%e A383178 .
%e A383178 0001 1001 2001 3001 0011 1011 2011 0021
%e A383178 0101 1101 2101 0111 1111
%e A383178 0201 1201
%e A383178 .
%e A383178 0002 1002 2002 0012
%e A383178 0102
%e A383178 .
%e A383178 0003
%t A383178 f[om_, lm_ : 0] := Block[{f, i, j, k, nn, w}, i = Abs[om]; j = 1;
%t A383178 If[lm == 0, nn = Times @@ Prime@ Range[i], nn = Abs[lm]]; w = ConstantArray[1, i];
%t A383178 Union@ Reap[Do[
%t A383178 While[Set[k, Times @@ Map[Prime, Accumulate@w]]; k <= nn,
%t A383178 If[Or[k == 1, Union[#2] == #1 - 1 & @@
%t A383178 TakeDrop[Map[Floor@Log[#, k] &, FactorInteger[k][[All, 1]] ], 1] ],
%t A383178 Sow[k]];
%t A383178 j = 1; w[[-j]]++];
%t A383178 If[j == i, Break[], j++; w[[-j]]++;
%t A383178 w = PadRight[w[[;; -j]], i, 1]], {n, Infinity}] ][[-1, 1]] ];
%t A383178 f[4, 2000000]
%Y A383178 Cf. A010846, A020639, A162306, A383177.
%K A383178 nonn
%O A383178 1,1
%A A383178 _Michael De Vlieger_, May 09 2025