cp's OEIS Frontend

A369295 a(n) is the first number k such that Omega(k) = n and Omega(n - 1) = Omega(n + 1) = n + 1.

%I A369295 #13 Feb 12 2024 01:50:29
%S A369295 5,51,343,3185,75951,1780624,16825375,212781249,4613781249,
%T A369295 74239460225,858245781249,46055847109375,487028345781249
%N A369295 a(n) is the first number k such that Omega(k) = n and Omega(n - 1) = Omega(n + 1) = n + 1.
%C A369295 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 A369295 From _David A. Corneth_, Jan 19 2024: (Start)
%C A369295 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 A369295 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 A369295 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 A369295 V:= Vector(8): count:= 0:
%p A369295 b:= 0: c:= 1:
%p A369295 for n from 3 do
%p A369295   a:= b; b:= c; c:= numtheory:-bigomega(n);
%p A369295   if a = b+1 and c = b+1 and b <= 8 and V[b] = 0 then
%p A369295     V[b]:= n-1; count:= count+1; if count = 8 then break fi;
%p A369295   fi
%p A369295 od:
%p A369295 convert(V,list);
%o A369295 (PARI)
%o A369295 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 A369295 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 A369295 Cf. A001222, A322300.
%K A369295 nonn,more
%O A369295 1,1
%A A369295 _Zak Seidov_ and _Robert Israel_, Jan 18 2024
%E A369295 a(9)-a(11) from _Daniel Suteu_, Jan 19 2024
%E A369295 a(12)-a(13) from _Martin Ehrenstein_, Feb 12 2024