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

1,2

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.

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.

Example

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

Crossrefs

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

Sequence in context: A133996 A284573 A355475 * A080742 A005417 A058215

Adjacent sequences: A371627 A371628 A371629 * A371631 A371632 A371633

Keyword

nonn,hard

Author

Michael De Vlieger, Jun 04 2024

Status

approved