A360316 a(n) is the smallest k such that k!'s prime(n)-smooth part is less than its prime(n+1)-rough part.

3, 21, 47, 111, 186, 293, 437, 619, 830, 1070, 1358, 1662, 2019, 2428, 2903, 3373, 3908, 4493, 5113, 5791, 6506, 7325, 8150, 9043, 9942, 10929, 11983, 13089, 14303, 15591, 16845, 18143, 19535, 21003, 22488, 24046, 25693, 27333, 29119, 30905, 32741, 34764, 36734

Offset

1,1

Links

Table of n, a(n) for n=1..43.

Example

For n=1, the prime(1)-smooth part of k! (i.e., the 2-smooth part) is the part whose prime factors are 2's; the prime(2)-rough part (i.e., the 3-rough part) is the odd part. For k = 1..3, we have
   k  k!  2-smooth part   3-rough part
   -  --  -------------   ------------
   1   1        1       =       1
   2   2        2       >       1
   3   6        2       <       3
so a(1)=3.
Similarly, for n=2, we have
   k  3-smooth part   5-rough part
  --  -------------   ------------
   1             1  =           1
   2             2  >           1
   3             6  >           1
   4            24  >           1
   5            24  >           5
   .             .  .           .
   .             .  .           .
  18     429981696  >    14889875
  19     429981696  >   282907625
  20    1719926784  >  1414538125
  21    5159780352  <  9901766875
so a(2)=21.

Crossrefs

Cf. A000142.

Sequence in context: A364051 A318549 A217000 * A034186 A318211 A219069

Adjacent sequences: A360313 A360314 A360315 * A360317 A360318 A360319

Keyword

nonn

Author

Jon E. Schoenfield, Feb 03 2023

Status

approved