A351166 a(1)=1; for n > 1, a(n) is the smallest number that has n divisors and is coprime to a(n-1).

1, 2, 9, 8, 81, 20, 729, 40, 441, 80, 59049, 140, 531441, 320, 3969, 440, 43046721, 700, 387420489, 560, 88209, 5120, 31381059609, 1120, 1185921, 20480, 53361, 4160, 22876792454961, 2800, 205891132094649, 3080, 9979281, 327680, 1750329, 8800, 150094635296999121

Offset

1,2

Comments

For every odd prime p, a(p) = 3^(p-1).

For n > 1, the smallest prime factor of a(n) is 2 + (n mod 2); see Examples.

For n > 4, a(n) is a multiple of 5 iff n is even.

Is 4 the largest composite m such that a(m) is a prime power?

Links

Table of n, a(n) for n=1..37.

Example

Given that a(27) = 53361 = 3^2 * 7^2 * 11^2, the prime factors of a(28) cannot include 3, 7, or 11, so the smallest primes available as prime factors of a(28) are 2, 5, and 13. The only prime signatures yielding 28 divisors are p^27, p^13 * q, p^6 * q^3, and p^6 * q * r, and the smallest candidate value with each of these signatures is 2^27 = 134217728, 2^13 * 5 = 40960, 2^6 * 5^3 = 8000, and 2^6 * 5 * 13 = 4160. The smallest of these is 4160, so a(28) = 4160. (This is the smallest term that is a multiple of 13.)
The table below lists the first several terms and their prime factorizations.
   n  a(n)
  --  ----
   1     1 = 1
   2     2 =   2^1
   3     9 =       3^2
   4     8 =   2^3
   5    81 =       3^4
   6    20 =   2^2  *  5^1
   7   729 =       3^6
   8    40 =   2^3  *  5^1
   9   441 =       3^2  *  7^2
  10    80 =   2^4  *  5^1

Crossrefs

Cf. A000005.

Sequence in context: A214300 A092397 A272348 * A092270 A249225 A191351

Adjacent sequences: A351163 A351164 A351165 * A351167 A351168 A351169

Keyword

nonn

Author

Jon E. Schoenfield, Mar 27 2022

Status

approved