A369228 - OEIS

%I #7 Jan 20 2024 09:25:34

%S 1,4,45,104,71874,392274,305778473,24405534712

%N a(n) is the least k that starts a sequence of exactly n numbers on which i + Omega(i) is constant, where Omega = A001222.

%C a(n) is the least k such that Omega(k + j) = Omega(k) - j for 0 <= j <= n-1, but not for j = -1 or j = n.

%C Dickson's conjecture implies that a(n) exists for all n.

%e Omega(1) = 0 while Omega(0) is undefined and Omega(2) = 1, so a(1) = 1.

%e Omega(4 .. 5) = (2, 1) while Omega(3) = 1 and Omega(6) = 2, so a(2) = 4.

%e Omega(45 .. 47) = (3, 2, 1) while Omega(44) = 3 and Omega(48) = 2  so a(3) = 45.

%e Omega(104 .. 107) = (4, 3, 2, 1) while Omega(103) = 1 and Omega(108) = 3 so a(4) = 104.

%e Omega(71874 .. 71878) = (7, 6, 5, 4, 3)  while Omega(71873) = 2 and Omega(71879) = 6 so a(5) = 71874.

%e Omega(392274 .. 392279) = (6, 5, 4, 3, 2, 1) while Omega(392273) = 2 and Omega(392280) = 5 so a(6) = 392274.

%e Omega(305778473 .. 305778479) = (7, 6, 5, 4, 3, 2, 1)  while Omega(305778472) = 4 and Omega(305778480) = 6 so a(7) = 305778473.

%p V:= Vector(7): count:= 0: v:= 1 + numtheory:-bigomega(1); u:= 1;

%p for i from 2 while count < 7 do

%p     w:= i + numtheory:-bigomega(i);

%p     if w <> v then

%p       if V[i-u] = 0 then V[i-u]:= u; count:= count+1 fi;

%p       u:= i; v:= w;

%p     fi;

%p od:

%p convert(V,list);

%Y Cf. A001222, A369139, A369185.

%K nonn,more

%O 1,2

%A _Zak Seidov_ and _Robert Israel_, Jan 16 2024

%E a(8) from _Daniel Suteu_, Jan 18 2024