A383594 a(0) = 0 and thereafter a(n) = 2 if a(n-1) is an odd prime, otherwise a(n) = a(n-1) + k where k = n - P(n) and P(n) is the number of odd primes among terms a(0),...,a(n-1).

0, 1, 3, 2, 5, 2, 6, 11, 2, 8, 15, 23, 2, 11, 2, 12, 23, 2, 14, 27, 41, 2, 17, 2, 18, 35, 53, 2, 21, 41, 2, 23, 2, 24, 47, 2, 26, 51, 77, 104, 132, 161, 191, 2, 33, 65, 98, 132, 167, 2, 38, 75, 113, 2, 41, 2, 42, 83, 2, 44, 87, 131, 2, 47, 2, 48, 95, 143, 192, 242

Offset: 0

Author

Aaron Pieniozek, May 01 2025

Keywords

• nonn

Comments

The number of terms between consecutive 2's is never 4*d, since the sum of 4*d consecutive values of k is always even and consequently a(n+4*d) is even (not a prime).

Examples

			The sequence, and the k increments applied, begin
  a(n) = 0, 1, 3, 2, 5, 2, 6, 11, 2, 8, 15, 23, ...
  k    =   1  2     3     4  5      6  7   8
		

Links

• Aaron Pieniozek, Table of n, a(n) for n = 0..199

Programs

Maple

seq := proc(n)
    local a, i, k;
    a := [0];
    k := 1;
    for i from 1 to n-1 do
        if isprime(a[-1]) and a[-1] <> 2 then
            a := [op(a), 2];
        else
            a := [op(a), a[-1] + k];
            k := k + 1;
        end if;
    end do;
    return a;
end proc:

Mathematica

sequence[n_] := Module[{a = {0}, k = 1},
  While[Length[a] < n,
    If[PrimeQ[Last[a]] && Last[a] != 2,
      AppendTo[a, 2],
      AppendTo[a, Last[a] + k];
      k++
    ];
  ];
  a
]