A393342 a(n) is the smallest prime number p, such that the sequence of n consecutive prime numbers modulo n starting with p is a palindromic.

2, 3, 2, 5, 347, 113, 3331, 1327, 8423, 26423, 1078967, 2887, 146520859, 26592563, 136963907, 588585923, 348325838381, 87793337

Offset

1,1

Comments

A block of odd length is called palindromic if the residues read the same from left to right and from right to left. For example, for q = 3, the block of 3 prime numbers [5, 7, 11] gives the residues modulo 3 [2, 1, 2], which is a palindrome.

Links

Table of n, a(n) for n=1..18.

Example

  n   a(n)           prime numbers                  modulo n
  ------------------------------------------------------------------
  1     2   [2]                                   [0]
  2     3   [3,5]                                 [1,1]
  3     2   [2,3,5]                               [2,0,2]
  4     5   [5,7,11,13]                           [1,3,3,1]
  5   347   [347,349,353,359,367]                 [2,4,3,4,2]|
  6   113   [113,127,131,137,139,149]             [5,1,5,5,5,1,5]
  7  3331   [3331,3343,3347,3359,3361,3371,3373]  [6,4,1,6,1,4,6]
  ...

Maple

FindLineU_fast := proc(n)
    local q, L, P, R, p, i, ok;
    q := n;
    L := n;
    P := [];
    R := [];
    p := 2;
    while true do
        P := [op(P), p];
        R := [op(R), p mod q];
        if nops(P) > L then
            P := P[2..-1];
            R := R[2..-1];
        end if;
        if nops(P) = L then
            ok := true;
            for i from 1 to n+1 do
                if R[i] <> R[L-i+1] then
                    ok := false;
                    break;
                end if;
            end do;
            if ok then
                return P[1], P, R;
            end if;
        end if;
        p := nextprime(p);
    end do;
end proc:
FindLineU_fast(3);

Prog

(PARI) ispal(v) = for (i=1, #v\2, if (v[i] != v[#v-i+1], return(0))); 1;
a(n) = my(v=Vecrev(primes(n)), p=prime(n)); v = apply(x->(x%n), v); while (!ispal(v), p=nextprime(p+1); v=concat(p % n, v); v=Vec(v, n)); for(i=1, n-1, p=precprime(p-1)); p; \\ Michel Marcus, Feb 21 2026
(Python)
from sympy import prime, nextprime
def A393342(n):
    p = [prime(i) for i in range(1, n+1)]
    r = [q%n for q in p]
    while True:
        if r[:(t:=len(r)+1>>1)]==r[:-t-1:-1]: return p[0]
        p = p[1:]+[m:=nextprime(p[-1])]
        r = r[1:]+[m%n] # Chai Wah Wu, Mar 03 2026

Crossrefs

Cf. A000040, A039701, A039703, A039705, A038194, A039709, A039711, A039713, A393417.

Sequence in context: A078599 A183103 A183105 * A316608 A033031 A134060

Adjacent sequences: A393339 A393340 A393341 * A393343 A393344 A393345

Keyword

nonn,hard,more

Author

Michel Lagneau, Feb 12 2026

Extensions

a(16) from Sean A. Irvine, Feb 25 2026

a(17)-a(18) from Chai Wah Wu, Mar 04 2026

Status

approved