A352127 a(n) is the smallest even number whose n smallest divisors > 2 are all odd.

2, 6, 30, 2002, 92378, 13357342, 2697562774, 292157776958, 36257787561098, 5563815981553006, 406158566653369438, 757411721425002359054, 78013407306775242982562, 17167365460734333186898502, 1939912297062979650119530726, 2548800423984304257055702381202

Offset

0,1

Comments

For n > 1, if the set of divisors of an even number m begins with 1, 2, and n odd divisors, then m must be divisible by 2 but not by 4, and its smallest odd divisor > 1 must be a prime p such that m has at least n odd divisors in the interval [p, 2*p-1] (since 2*p will be an even divisor), all of which must be prime (since, if any were composite, then p would not be m's smallest divisor > 1). Thus, the smallest such m is twice the product of the first run of n consecutive primes, the largest and smallest of which have a ratio less than 2.

Links

Table of n, a(n) for n=0..15.

Formula

a(n) = 2*Product_{j=k+1..k+n} prime(j) where k is the smallest integer such that 2*prime(k+1) > prime(k+n).

Example

The terms and their factorizations begin:
  a(0)  =                  2 = 2
  a(1)  =                  6 = 2 *  3
  a(2)  =                 30 = 2 *  3* 5
  a(3)  =               2002 = 2 *  7*11*13
  a(4)  =              92378 = 2 * 11*13*17*19
  a(5)  =           13357342 = 2 * 17*19*23*29*31
  a(6)  =         2697562774 = 2 * 23*29*31*37*41*43
  a(7)  =       292157776958 = 2 * 29*31*37*41*43*47*53
  a(8)  =     36257787561098 = 2 * 31*37*41*43*47*53*59*61
  a(9)  =   5563815981553006 = 2 * 37*41*43*47*53*59*61*67*71
  a(10) = 406158566653369438 = 2 * 37*41*43*47*53*59*61*67*71*73
  ...
a(n-1) divides a(n) whenever a(n-1) and a(n) have the same smallest odd prime factor; this happens at n = 2, 10, 12, 14, 19, 20, ..., which are the indices at which the largest prime factor of a(n) is less than twice the smallest prime factor of a(n-1). E.g., both a(9) and a(10) have 37 as their smallest odd prime factor, and 73 (the largest prime factor of a(10)) < 74 = 2*37.

Crossrefs

Cf. A080359.

Sequence in context: A102927 A227105 A127295 * A153046 A088260 A232171

Adjacent sequences: A352124 A352125 A352126 * A352128 A352129 A352130

Keyword

nonn

Author

Jon E. Schoenfield, Mar 05 2022

Status

approved