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A380475 a(n) is the least term in A380468 that has exactly n prime factors.

%I A380475 #38 Feb 19 2025 10:25:41
%S A380475 1,2,6,186,4686
%N A380475 a(n) is the least term in A380468 that has exactly n prime factors.
%C A380475 If it exists, a(5) > 2^35. - _Antti Karttunen_, Feb 19 2025
%C A380475 Conjecture: Sequence is finite and a(4) is the last term. This is equivalent to the claim that no arithmetic derivative (A003415) of a product of five or more distinct primes, (i.e., the value of the (n-1)-st elementary symmetric polynomial formed from those n distinct primes) can be formed as a carry-free sum of those n summands in primorial base (A049345). See also A380476 and A380528, A380530.
%H A380475 Wikipedia, <a href="https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial">Elementary symmetric polynomial</a>
%H A380475 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>.
%F A380475 a(n) = Min_{k in A380468} for which A001221(k) = n.
%e A380475 186 = 2*3*31 and A276086(186/2) = 2058 = 2 * 3 * 7^3, A276086(186/3) = 3 * 7^2, A276086(186/31) = 5, whose product =  2^1 * 3^2 * 5^1 * 7^5 = 1512630 = A380459(186), and as all the exponents are less than the corresponding primes, the product is in A048103, and because there are no any smaller number with three prime factors satisfying the same condition (of A380468), 186 is the term a(3) of this sequence. Note that A049345(A003415(186)) = 5121, where the digits are the exponents in the product read from the largest to the smallest prime factor.
%e A380475 See also the example in A380476 about 4686.
%Y A380475 Cf. A001221, A001222, A003415, A048103, A049345, A276086, A380459, A380468, A380476 (terms of A380468 with more than three prime factors).
%Y A380475 Cf. A047247, A047257.
%Y A380475 Cf. also A317836, A358235, A380525, A380528, A380530.
%K A380475 nonn,hard,more
%O A380475 0,2
%A A380475 _Antti Karttunen_, Feb 03 2025