A383178 -id:A383178 - cp's OEIS frontend

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

Michael De Vlieger, May 09 2025

nonn

A010846(a(n)) >= 176.

			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

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.

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

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]

A384000 Smallest number k with n distinct prime factors such that A010846(k) = A024718(n) (a tight lower bound), or -1 if such k does not exist.

Original entry on oeis.org

1, 2, 6, 1001, 268801, 3433936673, 2603508937756211

Offset: 0

Michael De Vlieger, May 19 2025

These numbers k have the smallest A010846(k) for a number with n distinct prime factors.
a(7) <= 1398483454696343742813089 = 1049 * 2819 * 3319 * 3433 * 3457 * 3463 * 3467.
a(8) <= 32829974457045619959776094471833047127947.

			Table of a(n), n = 0..6, showing prime decomposition and cardinality of row a(n) of A162306, c(n) = A010846(a(n)) = A024718(n).
n               a(n)   c(n)    prime factors of a(n)        a(n)
----------------------------------------------------------------------
0                  1     1     -
1                  2     2     2                            A000040(1)
2                  6     5     2,   3                       A138109(1)
3               1001    15     7,  11,  13                  A383177(1)
4             268801    50    13,  23,  29,  31             A383178(2)
5         3433936673   176    41,  83,  97, 101, 103        A383179(209)
6   2603508937756211   638   163, 373, 439, 457, 461, 463
Tables of terms m in r(a(n)) = row a(n) of A162306, writing instead only exponents i of prime power factors p^i | m for  each p | a(n), written in order of the prime base:
For n = 2, i.e., squarefree semiprime k in A138109 (that achieves the lower bound), we have the following ordered exponent combinations in a rank-2 table:
  00  10  20
  01  11
Thus row 6 of A162306 has the following elements:
   1   2   4
   3   6
For n = 3, i.e., sphenic k in A383177 (that achieves the lower bound), we have the following ordered exponent combinations in a rank-3 table:
  000 100 200 300     001 101 201     002
  010 110 210         011 111
  020 120
Thus row 1001 of A162306 has the following elements:
    1   7  49 343      13   91 637    169
   11  77 539         141 1001
  121 857

Cf. A001700, A001221, A005117, A007947, A010846, A024718, A138109, A162306, A383177, A383178, A383179.

cp's OEIS Frontend

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).

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