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

1,2

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)

Links

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

Formula

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

Example

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)

Mathematica

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

Crossrefs

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

Sequence in context: A328426 A176901 A304316 * A186415 A211038 A203264

Adjacent sequences: A377353 A377354 A377355 * A377357 A377358 A377359

Keyword

nonn

Author

David James Sycamore, Oct 26 2024

Extensions

More terms from Michael De Vlieger, Oct 26 2024

Status

approved