A369228 a(n) is the least k that starts a sequence of exactly n numbers on which i + Omega(i) is constant, where Omega = A001222.

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

Offset: 1

Author

Zak Seidov and Robert Israel, Jan 16 2024

Keywords

• nonn
• more

Comments

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.

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

Examples

			Omega(1) = 0 while Omega(0) is undefined and Omega(2) = 1, so a(1) = 1.
Omega(4 .. 5) = (2, 1) while Omega(3) = 1 and Omega(6) = 2, so a(2) = 4.
Omega(45 .. 47) = (3, 2, 1) while Omega(44) = 3 and Omega(48) = 2  so a(3) = 45.
Omega(104 .. 107) = (4, 3, 2, 1) while Omega(103) = 1 and Omega(108) = 3 so a(4) = 104.
Omega(71874 .. 71878) = (7, 6, 5, 4, 3)  while Omega(71873) = 2 and Omega(71879) = 6 so a(5) = 71874.
Omega(392274 .. 392279) = (6, 5, 4, 3, 2, 1) while Omega(392273) = 2 and Omega(392280) = 5 so a(6) = 392274.
Omega(305778473 .. 305778479) = (7, 6, 5, 4, 3, 2, 1)  while Omega(305778472) = 4 and Omega(305778480) = 6 so a(7) = 305778473.
		

Crossrefs

Cf. A001222, A369139, A369185.

Programs

Maple

V:= Vector(7): count:= 0: v:= 1 + numtheory:-bigomega(1); u:= 1;
for i from 2 while count < 7 do
    w:= i + numtheory:-bigomega(i);
    if w <> v then
      if V[i-u] = 0 then V[i-u]:= u; count:= count+1 fi;
      u:= i; v:= w;
    fi;
od:
convert(V,list);

Extensions

a(8) from Daniel Suteu, Jan 18 2024