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 A383177 #16 Jun 07 2025 11:58:51
%S A383177 1001,1309,1547,1729,2093,2261,3553,4199,4301,4807,5681,6061,6479,
%T A383177 7337,7843,8671,9269,9361,9889,10373,10879,11063,11339,11687,11803,
%U A383177 11891,12121,12617,13079,13717,13949,13981,14911,15283,15457,16211,16523,17081,17329,17719
%N A383177 Sphenic numbers k such that floor(log(k)/log(lpf(k))) = 1+floor(log(k)/log(p)) for all primes p | k such that p > lpf(k), where lpf = A020639(k).
%C A383177 Subset of A382022, a subset of A007304.
%C A383177 Let primes p, q, r, p < q < r divide k.
%C A383177 Then floor(log(k)/log(p)) = 3 and floor(log(k)/log(q)) = floor(log(k)/log(r)) = 2.
%C A383177 Row a(n) of A162306 is the set {1, p, p^2, p^3, q, p*q, p^2*q, q^2, p*q^2, r, p*r, p^2*r, q*r, p*q*r, r^2}.
%H A383177 Michael De Vlieger, <a href="/A383177/b383177.txt">Table of n, a(n) for n = 1..10000</a>
%H A383177 Michael De Vlieger, <a href="/A383177/a383177.png">Hasse diagram of R(1001)</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 A383177 Michael De Vlieger, <a href="/A383177/a383177_1.png">Three dimensional diagram of R(a(n))</a>, labeling exponents along axes, showing p^3, q^2, and r^2, and using the color scheme above.
%H A383177 Michael De Vlieger, <a href="/A383177/a383177_2.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.
%F A383177 A010846(a(n)) = 15.
%e A383177 Let s(n) = A010846(a(n)).
%e A383177 Table of a(n) for n = 1..12, showing prime factors of a(n) and
%e A383177 n a(n) facs(a(n)) s(n)
%e A383177 ---------------------------
%e A383177 1 1001 7*11*13 15
%e A383177 2 1309 7*11*17 15
%e A383177 3 1547 7*13*17 15
%e A383177 4 1729 7*13*19 15
%e A383177 5 2093 7*13*23 15
%e A383177 6 2261 7*17*19 15
%e A383177 7 3553 11*17*19 15
%e A383177 8 4199 13*17*19 15
%e A383177 9 4301 11*17*23 15
%e A383177 10 4807 11*19*23 15
%e A383177 11 5681 13*19*23 15
%e A383177 12 6061 11*19*29 15
%e A383177 Let f(p,k) = floor(log(k)/log(p)) and let w be the list of f(p,k) across the sorted list of distinct prime factors of k.
%e A383177 30 = 2*3*5 is not in the sequence since f(30,2) = 4, f(30,3) = 3, f(30,5) = 2.
%e A383177 a(1) = 1001 = 7*11*13; f(7,1001) = 3, f(11,1001) = 2, f(13,1001) = 2.
%e A383177 a(2) = 1309 = 7*11*17; w(1309) = {3,2,2}, etc.
%e A383177 Pattern of numbers in row a(n) of A275280:
%e A383177 Level r^0 Level r^1 Level r^2
%e A383177 1, p, p^2, p^3 | r, p*r, p^2*r | r^2
%e A383177 q, p*q, p^2*q | q*r, p*q*r |
%e A383177 q^2, p*q^2; |
%e A383177 Example: k = 1001 = 7*11*13
%e A383177 1, 7, 49, 343 | 13, 91, 637 | 169
%e A383177 11, 77, 539 | 143, 1001 |
%e A383177 121, 847 |
%t A383177 f[om_, lm_ : 0] := Block[{f, i, j, k, nn, w}, i = Abs[om]; j = 1;
%t A383177 If[lm == 0, nn = Times @@ Prime@ Range[i], nn = Abs[lm]]; w = ConstantArray[1, i];
%t A383177 Union@ Reap[Do[
%t A383177 While[Set[k, Times @@ Map[Prime, Accumulate@w]]; k <= nn,
%t A383177 If[Or[k == 1, Union[#2] == #1 - 1 & @@
%t A383177 TakeDrop[Map[Floor@Log[#, k] &, FactorInteger[k][[All, 1]] ], 1] ],
%t A383177 Sow[k]];
%t A383177 j = 1; w[[-j]]++];
%t A383177 If[j == i, Break[], j++; w[[-j]]++;
%t A383177 w = PadRight[w[[;; -j]], i, 1]], {n, Infinity}] ][[-1, 1]] ];
%t A383177 f[3, 20000]
%Y A383177 Cf. A005117, A007304, A010846, A162306, A381250.
%Y A383177 Intersection of A380995 and A382022.
%K A383177 nonn
%O A383177 1,1
%A A383177 _Michael De Vlieger_, Apr 21 2025