A371630 Numbers k that set records in A372720.

1, 2, 6, 12, 30, 60, 120, 210, 420, 840, 1260, 1680, 2520, 4620, 9240, 13860, 18480, 27720, 32760, 55440, 65520, 102960, 110880, 120120, 180180, 240240, 360360, 556920, 720720, 1081080, 1441440, 1884960, 2162160, 2827440, 2882880, 3063060, 3603600, 4084080, 6126120

Offset: 1

Author

Michael De Vlieger, Jun 04 2024

Keywords

• nonn
• hard

Comments

In other words, numbers k that set records for d(k) - f(k), where d = A000005 and f = A008479.

Largest primorial in this sequence is A002110(4) = 210.

The primorials A002110(0..4) are the only squarefree numbers in this sequence.

{a(n)} \ A002110(0..4) is contained in A126706.

Not a subset of A060735; a(13) = 2520 is not in A060735. Though common for small n, the set of a(n) in A060735 is likely finite; the restriction is connected with the finite number of primorials in the sequence.

Not a subset of A025487 or A055932; a(19) = 32760 is the smallest term without a primorial kernel.

The prime shape of a(n) appears to feature exponents m of prime power factors p^m | a(n) that are nonincreasing as pi(p) increases.

Examples

			Table of a(n) and A371634(n) = b(n) for n = 1..20. Asterisks in the a(n) column denote squarefree terms while "+" denotes numbers not in A055932 (i.e., in A080259).
   n     a(n)  A067255(a(n))            d(n)-f(n) = b(n)
  ------------------------------------------------------
   1       1*  1                          1 -  1 =   0
   2       2*  2                          2 -  1 =   1
   3       6*  2 * 3                      4 -  1 =   3
   4      12   2^2 * 3                    6 -  2 =   4
   5      30*  2 * 3 * 5                  8 -  1 =   7
   6      60   2^2 * 3 * 5               12 -  2 =  10
   7     120   2^3 * 3 * 5               16 -  4 =  12
   8     210*  2 * 3 * 5 * 7             16 -  1 =  15
   9     420   2^2 * 3 * 5 * 7           24 -  2 =  22
  10     840   2^3 * 3 * 5 * 7           32 -  4 =  28
  11    1260   2^2 * 3^2 * 5 * 7         36 -  6 =  30
  12    1680   2^4 * 3 * 5 * 7           40 -  8 =  32
  13    2520   2^3 * 3^2 * 5 * 7         48 - 11 =  37
  14    4620   2^2 * 3 * 5 * 7 * 11      48 -  2 =  46
  15    9240   2^3 * 3 * 5 * 7 * 11      64 -  4 =  60
  16   13860   2^2 * 3^2 * 5 * 7 * 11    72 -  6 =  66
  17   18480   2^4 * 3 * 5 * 7 * 11      80 -  8 =  72
  18   27720   2^3 * 3^2 * 5 * 7 * 11    96 - 12 =  84
  19   32760+  2^3 * 3^2 * 5 * 7 * 13    96 - 11 =  85
  20   55440   2^4 * 3^2 * 5 * 7 * 11   120 - 20 = 100
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..67 (terms 1..56 from Michael De Vlieger)
• Michael De Vlieger, Prime power decomposition of A371630(n), n = 1..56.

Crossrefs

Cf. A000005, A002110, A008479, A055932, A060735, A080259, A126706, A371634, A372720.