A354400 Replace the nonprimes in the prime gaps with primes. See Comments section for details.

2, 3, 5, 11, 11, 19, 17, 29, 41, 29, 59, 63, 41, 77, 95, 113, 59, 141, 129, 71, 173, 161, 203, 225, 203, 101, 221, 107, 231, 311, 269, 335, 137, 375, 149, 391, 417, 357, 455, 473, 179, 525, 191, 411, 197, 585, 645, 485, 227, 503, 645, 239, 741, 699, 729, 755, 269, 783

Offset

1,1

Comments

Start with the sequence of the nonprime numbers. For visual clarity, it is interrupted with spaces where the primes are missing, and the nonprimes or their groups in the prime gaps are distinct:

1 4 6 8 9 10 12 14 15 16 18 20 21 22 ...

Replace the first nonprime 1 with the first prime 2 and thereafter every single or every first nonprime in the prime gaps with successive primes:

2 3 5 7 9 10 11 13 15 16 17 19 21 22 ...

Next, replace every second nonprime in the prime gaps with a new sequence of the primes:

2 3 5 7 2 10 11 13 3 16 17 19 5 22 ...

Follow by replacing every third nonprime in the prime gaps with yet another prime sequence:

2 3 5 7 2 2 11 13 3 3 17 19 5 5 ...

And so on. The sums of the substituting primes in the individual prime gaps form the terms of the sequence:

2 3 5 11 11 19 17 29 ...

Links

Table of n, a(n) for n=1..58.

Prog

(MATLAB)
function a = A354400( max_prime )
    % fill the gaps
    p = primes(max_prime); b = [1:max_prime];
    b(isprime(b) == 0) = 0;
    pj = [0 find(b > 0)]; pjo = pj;
    j = pj(b(pj+1) == 0)+1;
    while ~isempty(j)
        b(j) = p(1:length(j));
        pj = find(b(1:end-1) > 0);
        if ~isempty(find(b == 0, 1))
            j = pj(b(pj+1) == 0)+1;
        else
            j = [];
        end
    end
    % sum the gaps
    k = 1;
    for n = 1:length(pjo)-1
        m = sum(b(pjo(n)+1:pjo(n+1)-1));
        if m > 0
            a(k) = m; k = k+1;
        end
    end
end % Thomas Scheuerle, May 25 2022

Crossrefs

Cf. A000040, A018252.

Sequence in context: A272197 A207982 A259155 * A210144 A243357 A066159

Adjacent sequences: A354397 A354398 A354399 * A354401 A354402 A354403

Keyword

nonn

Author

Tamas Sandor Nagy, May 25 2022

Status

approved