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A321590 Smallest number m that is a product of exactly n primes and is such that m-1 and m+1 are products of exactly n-1 primes.

%I A321590 #28 Jan 08 2019 03:07:55
%S A321590 4,50,189,1863,10449,447849,4449249,5745249,3606422049,16554218751,
%T A321590 105265530369,1957645712385
%N A321590 Smallest number m that is a product of exactly n primes and is such that m-1 and m+1 are products of exactly n-1 primes.
%C A321590 From _Jon E. Schoenfield_, Nov 15 2018: (Start)
%C A321590 If a(11) is odd, it is 16554218751.
%C A321590 If a(12) is odd, it is 105265530369.
%C A321590 If a(13) is odd, it is 1957645712385. (End)
%C A321590 a(11), a(12), and a(13) are indeed odd. - _Giovanni Resta_, Jan 04 2019
%C A321590 10^13 < a(14) <= 240455334218751, a(15) <= 2992278212890624. - _Giovanni Resta_, Jan 06 2019
%e A321590 For n = 3, 50 = 2*5*5, and the numbers before and after 50 are 49 = 7*7 and 51 = 3*17.
%t A321590 a[n_] := Module[{o={0,0,0}, k=1}, While[o!={n-1,n,n-1}, o=Rest[AppendTo[o,PrimeOmega[k]]]; k++]; k-2]; Array[a,7,2] (* _Amiram Eldar_, Nov 14 2018 *)
%o A321590 (PARI) {for(n=2,10,for(k=2^n,10^12,if(n==bigomega(k) &&
%o A321590 n-1==bigomega(k-1) && n-1==bigomega(k+1),print1(k", ");break())))}
%Y A321590 Cf. A078840.
%Y A321590 Sequences listing r-almost primes, that is, the n such that A001222(n) = r: A000040 (r = 1), A001358 (r = 2), A014612 (r = 3), A014613 (r = 4), A014614 (r = 5), A046306 (r = 6), A046308 (r = 7), A046310 (r = 8), A046312 (r = 9), A046314 (r = 10), A069272 (r = 11), A069273 (r = 12), A069274 (r = 13), A069275(r = 14), A069276 (r = 15), A069277 (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), A069281 (r = 20).
%K A321590 nonn,more
%O A321590 2,1
%A A321590 _Zak Seidov_, Nov 13 2018
%E A321590 a(10) from _Jon E. Schoenfield_, Nov 14 2018
%E A321590 a(11)-a(13) from _Giovanni Resta_, Jan 04 2019