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%I #4 Mar 05 2022 23:02:29
%S 2,6,30,2002,92378,13357342,2697562774,292157776958,36257787561098,
%T 5563815981553006,406158566653369438,757411721425002359054,
%U 78013407306775242982562,17167365460734333186898502,1939912297062979650119530726,2548800423984304257055702381202
%N a(n) is the smallest even number whose n smallest divisors > 2 are all odd.
%C 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.
%F 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).
%e The terms and their factorizations begin:
%e a(0) = 2 = 2
%e a(1) = 6 = 2 * 3
%e a(2) = 30 = 2 * 3* 5
%e a(3) = 2002 = 2 * 7*11*13
%e a(4) = 92378 = 2 * 11*13*17*19
%e a(5) = 13357342 = 2 * 17*19*23*29*31
%e a(6) = 2697562774 = 2 * 23*29*31*37*41*43
%e a(7) = 292157776958 = 2 * 29*31*37*41*43*47*53
%e a(8) = 36257787561098 = 2 * 31*37*41*43*47*53*59*61
%e a(9) = 5563815981553006 = 2 * 37*41*43*47*53*59*61*67*71
%e a(10) = 406158566653369438 = 2 * 37*41*43*47*53*59*61*67*71*73
%e ...
%e 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.
%Y Cf. A080359.
%K nonn
%O 0,1
%A _Jon E. Schoenfield_, Mar 05 2022