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%I #13 Feb 12 2024 01:50:29
%S 5,51,343,3185,75951,1780624,16825375,212781249,4613781249,
%T 74239460225,858245781249,46055847109375,487028345781249
%N a(n) is the first number k such that Omega(k) = n and Omega(n - 1) = Omega(n + 1) = n + 1.
%C a(n) is the first k such that k - 1, k and k + 1 have n + 1, n and n + 1 prime factors respectively, counted with multiplicity.
%C From _David A. Corneth_, Jan 19 2024: (Start)
%C a(9) <= 4613781249. This value is exact if 27000 divides lcm(a(9)-1, a(9), a(9)+1). Same goes for a(10) <= 104168181249.
%C a(11) <= 858245781249 if 810000 divides lcm(a(11)-1, a(11), a(11)+1). a(12) <= 46055847109375 if 24300000 divides lcm(a(12)-1, a(12), a(12)+1). (End)
%e a(3) = 343 because 343 = 7^3 has 3 prime factors while 342 = 2 * 3^2 * 19 and 344 = 2^3 * 43 each have 4, and 343 is the least number that works.
%p V:= Vector(8): count:= 0:
%p b:= 0: c:= 1:
%p for n from 3 do
%p a:= b; b:= c; c:= numtheory:-bigomega(n);
%p if a = b+1 and c = b+1 and b <= 8 and V[b] = 0 then
%p V[b]:= n-1; count:= count+1; if count = 8 then break fi;
%p fi
%p od:
%p convert(V,list);
%o (PARI)
%o generate(A, B, n) = A=max(A, 2^n); (f(m, p, k) = my(list=List()); if(k==1, forprime(q=max(p, ceil(A/m)), B\m, my(t=m*q); if(bigomega(t+2) == n && bigomega(t+1) == n-1, listput(list, t+1))), forprime(q = p, sqrtnint(B\m, k), list=concat(list, f(m*q, q, k-1)))); list); vecsort(Vec(f(1, 2, n)));
%o a(n) = my(x=2^n, y=2*x); while(1, my(v=generate(x, y, n+1)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ _Daniel Suteu_, Jan 19 2024
%Y Cf. A001222, A322300.
%K nonn,more
%O 1,1
%A _Zak Seidov_ and _Robert Israel_, Jan 18 2024
%E a(9)-a(11) from _Daniel Suteu_, Jan 19 2024
%E a(12)-a(13) from _Martin Ehrenstein_, Feb 12 2024
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