A393703 - OEIS

%I #38 Mar 06 2026 08:08:15

%S 1,2,5,3,4,41,3,40,11425,4417937,2668278,4417936,162041584

%N a(n) is the smallest index k such that the subsequence b(k), b(k+1),..., b(k+n-1) is a palindrome of length n, where b(k) = A206400(k).

%C The aim of this sequence is to study the local symmetries of the differences between prime numbers of the form m^2+1. To do this, we identify palindromes in subsequences of A206400. This research falls within the framework of the fourth problem posed by the mathematician Edmund Landau at the 1912 International Congress of Mathematicians (Landau, 1912). This problem, still unsolved to this day, conjectures the existence of infinitely many prime numbers of the form k^2+1.

%C While classical mathematical research attempts to prove the infinitude of these numbers, we have chosen to explore the internal structure of their distribution through the sequence A206400. The objective of this sequence is to identify the emergence of palindromic subsequences within A206400.

%H Michel Lagneau, <a href="/A393703/a393703_1.txt">Maple program</a>

%H Edmund Landau, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002123789">Gelöste und ungelöste Probleme aus der Theorie der Primzahlverteilung und der Riemannschen Zetafunktion</a>, Jahresbericht der Deutschen Mathematiker-Vereinigung (1912), Vol. 21, page 208-228, here p. 224.

%H János Pintz, <a href="https://doi.org/10.5802/jtnb.676">Landau's problems on primes</a>, Journal de Théorie des Nombres de Bordeaux, vol. 21, no. 2, pp. 357-404 (2009).

%e Initial solutions:

%e    n      a(n)  palindrome

%e   --------------------------------------------

%e    1        1   [0]

%e    2        2   [1, 1]

%e    3        5   [3, 1, 3]

%e    4        3   [1, 3, 3, 1]

%e    5        4   [3, 3, 1, 3, 3]

%e    6       41   [9, 5, 3, 3, 5, 9]

%e    7        3   [1, 3, 3, 1, 3, 3, 1]

%e    8       40   [3, 9, 5, 3, 3, 5, 9, 3]

%e    9    11425   [7, 1, 13, 3, 11, 3, 13, 1, 7]

%e   10  4417937   [9, 5, 37, 21, 3, 3, 21, 37, 5, 9]

%e   11  2668278   [1, 49, 7, 15, 45, 57, 45, 15, 7, 49, 1]

%e   12  4417936   [5, 9, 5, 37, 21, 3, 3, 21, 37, 5, 9, 5]

%p # see linked program

%Y Cf. A206400.

%K nonn,more

%O 1,2

%A _Michel Lagneau_, Feb 25 2026

%E a(9) corrected and a(10)-a(13) from _Sean A. Irvine_, Mar 04 2026