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).

1001, 1309, 1547, 1729, 2093, 2261, 3553, 4199, 4301, 4807, 5681, 6061, 6479, 7337, 7843, 8671, 9269, 9361, 9889, 10373, 10879, 11063, 11339, 11687, 11803, 11891, 12121, 12617, 13079, 13717, 13949, 13981, 14911, 15283, 15457, 16211, 16523, 17081, 17329, 17719

Offset: 1

Author

Michael De Vlieger, Apr 21 2025

Keywords

• nonn

Comments

Subset of A382022, a subset of A007304.

Let primes p, q, r, p < q < r divide k.

Then floor(log(k)/log(p)) = 3 and floor(log(k)/log(q)) = floor(log(k)/log(r)) = 2.

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}.

Examples

			Let s(n) = A010846(a(n)).
Table of a(n) for n = 1..12, showing prime factors of a(n) and
 n   a(n)  facs(a(n))  s(n)
---------------------------
 1   1001    7*11*13    15
 2   1309    7*11*17    15
 3   1547    7*13*17    15
 4   1729    7*13*19    15
 5   2093    7*13*23    15
 6   2261    7*17*19    15
 7   3553   11*17*19    15
 8   4199   13*17*19    15
 9   4301   11*17*23    15
10   4807   11*19*23    15
11   5681   13*19*23    15
12   6061   11*19*29    15
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.
30 = 2*3*5 is not in the sequence since f(30,2) = 4, f(30,3) = 3, f(30,5) = 2.
a(1) = 1001 = 7*11*13; f(7,1001) = 3, f(11,1001) = 2, f(13,1001) = 2.
a(2) = 1309 = 7*11*17; w(1309) = {3,2,2}, etc.
Pattern of numbers in row a(n) of A275280:
  Level r^0                    Level r^1               Level r^2
  1,   p,     p^2,  p^3   |    r,   p*r,   p^2*r   |   r^2
  q,   p*q,   p^2*q       |    q*r, p*q*r          |
  q^2, p*q^2;             |
Example: k = 1001 = 7*11*13
    1,   7,  49, 343   |    13,   91, 637   |   169
   11,  77, 539        |   143, 1001        |
  121, 847             |
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Hasse diagram of R(1001) 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.
• Michael De Vlieger, Three dimensional diagram of R(a(n)), labeling exponents along axes, showing p^3, q^2, and r^2, and using the color scheme above.
• 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. A005117, A007304, A010846, A162306, A381250.

Intersection of A380995 and A382022.

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[3, 20000]

Formula

A010846(a(n)) = 15.