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 A352127 a(n) is the smallest even number whose n smallest divisors > 2 are all odd. 0
 2, 6, 30, 2002, 92378, 13357342, 2697562774, 292157776958, 36257787561098, 5563815981553006, 406158566653369438, 757411721425002359054, 78013407306775242982562, 17167365460734333186898502, 1939912297062979650119530726, 2548800423984304257055702381202 (list; graph; refs; listen; history; text; internal format)
 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 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

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Last modified April 1 14:50 EDT 2023. Contains 361695 sequences. (Running on oeis4.)