A394423 a(n) is the smallest index k such that the subsequence of prime gaps b(k), b(k+1),..., b(k+2n) is a palindrome of length 2n+1, where b(k) = A001223(k).

3, 3, 7, 6, 33, 541133, 54271403, 80107607, 66142651075, 19180434388696, 625795910115891

Offset

1,1

Comments

All palindromes considered are odd-length.

This sequence illustrates the local symmetric structure of prime gaps and the frequent occurrence of the (6,6) pair in prime constellations.

Links

Table of n, a(n) for n=1..11.

Rémy Sigrist, C++ program

Formula

a(n) = primepi(A055382(n+1)) = primepi(A175309(2*n+1)). - Pontus von Brömssen, Apr 07 2026

Example

Initial solutions:
n  | a(n)=k | [A001223(k), A001223(k+1),...,A001223(k+2n)]
------------------------------------------------------------
1  |      3 | [2, 4, 2]
2  |      3 | [2, 4, 2, 4, 2]
3  |      7 | [2, 4, 6, 2, 6, 4, 2]
4  |      6 | [4, 2, 4, 6, 2, 6, 4, 2, 4]
5  |     33 | [2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2]
6  | 541133 | [4, 6, 12, 18, 2, 10, 20, 10, 2, 18, 12, 6, 4]

Maple

next_gap := proc(i)
ithprime(i+1) - ithprime(i);
end proc:
find_smallest_k_palindrome := proc(n, max_iter)
    local A, i, k, j, sublist;
    A := [];
    for i from 1 to max_iter do
        A := [op(A), next_gap(i)];
        if nops(A) >= 2*n+1 then
            for k from 1 to nops(A)-2*n do
                sublist := [seq(A[t], t=k..k+2*n)];
                if sublist = ListTools:-Reverse(sublist) then
                    return [k, sublist];
                end if;
            end do;
        end if;
    end do;
    return FAIL;
end proc:for n from 1 to 8 do
    result := find_smallest_k_palindrome(n, 20000);
    if result <> FAIL then
        printf("n=%d, k=%d, %a\n", n, result[1], result[2]);
    else
        printf("n=%d, aucun\n", n);
    end if;
end do:

Prog

(C++) // See Links section.

Crossrefs

Cf. A000005, A055382, A175309, A001223, A393833, A394059.

Sequence in context: A000199 A243099 A324877 * A380734 A359947 A201932

Adjacent sequences: A394420 A394421 A394422 * A394424 A394425 A394426

Keyword

nonn,hard,more

Author

Michel Lagneau, Mar 20 2026

Extensions

a(7)-a(8) from Sean A. Irvine, Mar 28 2026

a(9) from Rémy Sigrist, Mar 29 2026

a(10)-a(11) from Pontus von Brömssen, Apr 07 2026

Status

approved