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Numbers k that set records in A372720.
3

%I #30 Jun 14 2024 11:34:23

%S 1,2,6,12,30,60,120,210,420,840,1260,1680,2520,4620,9240,13860,18480,

%T 27720,32760,55440,65520,102960,110880,120120,180180,240240,360360,

%U 556920,720720,1081080,1441440,1884960,2162160,2827440,2882880,3063060,3603600,4084080,6126120

%N Numbers k that set records in A372720.

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

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

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

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

%C 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.

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

%C 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.

%H Michael S. Branicky, <a href="/A371630/b371630.txt">Table of n, a(n) for n = 1..67</a> (terms 1..56 from Michael De Vlieger)

%H Michael De Vlieger, <a href="/A371630/a371630.txt">Prime power decomposition of A371630(n)</a>, n = 1..56.

%e 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).

%e n a(n) A067255(a(n)) d(n)-f(n) = b(n)

%e ------------------------------------------------------

%e 1 1* 1 1 - 1 = 0

%e 2 2* 2 2 - 1 = 1

%e 3 6* 2 * 3 4 - 1 = 3

%e 4 12 2^2 * 3 6 - 2 = 4

%e 5 30* 2 * 3 * 5 8 - 1 = 7

%e 6 60 2^2 * 3 * 5 12 - 2 = 10

%e 7 120 2^3 * 3 * 5 16 - 4 = 12

%e 8 210* 2 * 3 * 5 * 7 16 - 1 = 15

%e 9 420 2^2 * 3 * 5 * 7 24 - 2 = 22

%e 10 840 2^3 * 3 * 5 * 7 32 - 4 = 28

%e 11 1260 2^2 * 3^2 * 5 * 7 36 - 6 = 30

%e 12 1680 2^4 * 3 * 5 * 7 40 - 8 = 32

%e 13 2520 2^3 * 3^2 * 5 * 7 48 - 11 = 37

%e 14 4620 2^2 * 3 * 5 * 7 * 11 48 - 2 = 46

%e 15 9240 2^3 * 3 * 5 * 7 * 11 64 - 4 = 60

%e 16 13860 2^2 * 3^2 * 5 * 7 * 11 72 - 6 = 66

%e 17 18480 2^4 * 3 * 5 * 7 * 11 80 - 8 = 72

%e 18 27720 2^3 * 3^2 * 5 * 7 * 11 96 - 12 = 84

%e 19 32760+ 2^3 * 3^2 * 5 * 7 * 13 96 - 11 = 85

%e 20 55440 2^4 * 3^2 * 5 * 7 * 11 120 - 20 = 100

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

%K nonn,hard

%O 1,2

%A _Michael De Vlieger_, Jun 04 2024