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A309781 a(n) is the smallest even number m having n distinct odd prime divisors p_1, p_2, ..., p_n, each of which (p_i; i=1..n) has the property that there exists a k_i (0 < k_i < p_i-1) such that p_i - k_i | m - k_i.

%I A309781 #68 Oct 22 2019 04:23:39
%S A309781 20,110,2926,43010,704990,37461710,859382810,48530806610,
%T A309781 2383068532130,139761750534406,6586251483915290,302528651777276210,
%U A309781 37556939168033169170,2727217723862008961870,222939356264469226235810
%N A309781 a(n) is the smallest even number m having n distinct odd prime divisors p_1, p_2, ..., p_n, each of which (p_i; i=1..n) has the property that there exists a k_i (0 < k_i < p_i-1) such that p_i - k_i | m - k_i.
%C A309781 Subsequence of A309769 and A309780. a(1) is the only nonsquarefree known term. Conjecture: For n > 1 the minimal condition requires m to be squarefree and for every odd prime divisor p of m to be such that m/p - 1 is composite with least prime divisor q < p (k=p-q). No term is divisible by 3.
%C A309781 Not all terms are coprime to 7 and the terms aren't all 97-smooth. For n = 16..18, there are upper bounds on a(n): 16808251841257353520347590, 1627869069994521415245268370, 202089221222977079276742661490. - _David A. Corneth_, Sep 26 2019
%e A309781 a(2) = 110 = (2*5)*11; q = 3 < 11; also 110=(2*11)*5; q = 3 < 5.
%t A309781 kQ[n_, p_] := Module[{ans = False}, Do[If[Divisible[n - k, p - k], ans = True; Break[]], {k, 1, p - 2}]; ans]; aQ[n_] := EvenQ[n] && Length[(p = FactorInteger[ n][[2 ;; -1, 1]])] > 0 && AllTrue[p, kQ[n, #] &]; oddomega[n_] := PrimeNu[n / 2^IntegerExponent[n, 2]]; s = {}; om = 1; Do[If[oddomega[n] == om && aQ[n], AppendTo[s, n]; om++], {n, 2, 10^16, 2}]; s (* _Amiram Eldar_, Aug 17 2019 *)
%o A309781 (PARI) getk(p, m) = {for (k=1, p-2, if (((m-k) % (p-k)) == 0, return(k)); ); }
%o A309781 isok1(m) = {if ((m % 2) == 0, my(f = factor(m)[, 1]~); if (#f == 1, return (0)); for (i=2, #f, if (!getk(f[i], m), return(0)); ); return (1); ); }
%o A309781 isok(k, n) = (omega(k/(2^valuation(k, 2))) == n) && isok1(k);
%o A309781 a(n) = {my(k=2*prod(k=2, n+1, prime(k))); while (!isok(k, n), k+=2); k;} \\ _Michel Marcus_, Aug 27 2019
%Y A309781 Cf. A309769, A309780.
%K A309781 nonn,more
%O A309781 1,1
%A A309781 _David James Sycamore_, Aug 17 2019
%E A309781 a(8) from _Michel Marcus_, Sep 25 2019
%E A309781 a(9)-a(15) from _David A. Corneth_, Sep 26 2019