A377356 a(n) = Product{i = 1..(n-1)} prime(i)^e_i, where prime(i)^e_i is the smallest power of prime(i) which exceeds prime(n).

1, 4, 72, 1800, 529200, 64033200, 21643221600, 6254891042400, 2258015666306400, 17917354312141284000, 15068494976510819844000, 28961647344853795740168000, 39648495215104846368289992000, 66649120456591246745095476552000, 123234223724237215231681536144648000, 1905570801447880059127491593404692024000

Offset: 1

Author

David James Sycamore, Oct 26 2024

Keywords

• nonn

Comments

a(n) is the product of powers of primes p, for all p < prime(n), where each prime power is the smallest which exceeds prime(n), (compare with A099795). Every term may be expressed as a product of primorial powers, (A002110(n-1)^2 being the greatest primorial power divisor of a(n)).

From Michael De Vlieger, Oct 26 2024: (Start)

This sequence adds 1 to all exponents of prime power factors of A099795(n) for n > 1.

Proper subset of A001694, all terms are powerful. (End)

Examples

			For n = 5, a(5) = 529200, since prime(5) = 11, thus we have 2^4*3^3*5^2*7^2 = 16*27*25*49 = 529200. We may express this instead as 210*2520 = A002110(4)*A099795(5) = 210^2*6^1*2^1 = 529200.
From _Michael De Vlieger_, Oct 26 2024: (Start)
Table of first 12 terms showing exponents of prime power factors of a(n), where "." represents 0.
                                    Exponents of primes
                                            1 1 1 1 2 2 3
   n                          a(n)  2 3 5 7 1 3 7 9 3 9 1
  -------------------------------------------------------
   1                            1   . . . . . . . . . . .
   2                            4   2 . . . . . . . . . .
   3                           72   3 2 . . . . . . . . .
   4                         1800   3 2 2 . . . . . . . .
   5                       529200   4 3 2 2 . . . . . . .
   6                     64033200   4 3 2 2 2 . . . . . .
   7                  21643221600   5 3 2 2 2 2 . . . . .
   8                6254891042400   5 3 2 2 2 2 2 . . . .
   9             2258015666306400   5 3 2 2 2 2 2 2 . . .
  10         17917354312141284000   5 4 3 2 2 2 2 2 2 . .
  11      15068494976510819844000   5 4 3 2 2 2 2 2 2 2 .
  12   28961647344853795740168000   6 4 3 2 2 2 2 2 2 2 2 (End)
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..193

Crossrefs

Cf. A000040, A001694, A002110, A007947, A099795.

Programs

Mathematica

Array[Product[Prime[i]^(1 + Floor[Log[Prime[i], Prime[#]]]), {i, # - 1}] &, 12] (* Michael De Vlieger, Oct 26 2024 *)

Formula

a(n) = A002110(n-1)*A099795(n); A007947(a(n)) = rad(a(n)) = A002110(n-1).

Extensions

More terms from Michael De Vlieger, Oct 26 2024