A393417 a(n) = p is the smallest prime number, such that the sequence of 2n+1 consecutive prime numbers modulo n starting with p mod n is palindromic.

2, 3, 5, 41, 45667, 79, 1370321, 1114577, 184661011, 4178329, 1911129302867, 17213047

Offset

1,1

Comments

An estimate of the average position of length 2*n+1 is phi(n)^n.

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 n = 3, the block of 7 prime numbers [5,7,11,13,17] gives the residues modulo 3 [2,1,2,1,2,1,2], which is a palindrome.

Links

Table of n, a(n) for n=1..12.

Example

*---*------*-------------*--------------------------------------*
| n |    k |   prime(k)  | [prime(k), ..., prime(k+2n)] (mod n) |
*---*------*-------------*--------------------------------------*
| 1 |    1 |           2 | [0,0,0] (mod 1)                      |
| 2 |    2 |           3 | [1,1,1,1,1] (mod 2)                  |
| 3 |    3 |           5 | [2,1,2,1,2,1,2] (mod 3)              |
| 4 |   13 |          41 | [1,3,3,1,3,1,3,3,1] (mod 4)          |
| 5 | 4732 |       45667 | [2,3 2,1,2,2,2,1,2,3,2] (mod 5)      |
| 6 |   22 |          79 | [1,5,5,1,5,1,5,1,5,1,5,5,1] (mod 6)  |

Maple

# Function to find the first palindrome modulo n
#  if the program prints "FAIL", increase the maximum value kmax.
with(ListTools):
FindLineU_simple := proc(n, P, kmax)
    local R, k, i, ok;
    R := [seq(P[i] mod n, i=1..kmax+2*n)];
     for k from 1 to kmax do
        ok := true;
        for i from 0 to n do
            if R[k+i] <> R[k+2*n-i] then
                ok := false;
                break;
            end if;
        end do;
        if ok then
            return k, [seq(R[k+i], i=0..2*n)];
        end if;
    end do;
    return FAIL;
end proc:
# Display for n=3n=1..8
TrianglePal_1to8 := proc(kmax)
    local P, n, res;
    # Generate enough prime numbers
    P := [seq(ithprime(i), i=1..kmax+16)];  # 2*8=16 pour le plus grand palindrome
    for n from 1 to 8 do
        res := FindLineU_simple(n, P, kmax);
        print(n, res);
    end do;
end proc:TrianglePal_1to8(10^5);

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(2*n+1)), p=prime(2*n+1)); v = apply(x->(x%n), v); while (!ispal(v), p=nextprime(p+1); v=concat(p % n, v); v=Vec(v, 2*n+1)); for(i=1, 2*n, p=precprime(p-1)); p; \\ Michel Marcus, Feb 15 2026
(Python)
from sympy import prime, nextprime
def A393417(n):
    p = [prime(i) for i in range(1, n+1<<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, Feb 20 2026

Crossrefs

Cf. A393342.

Sequence in context: A136015 A106713 A106820 * A362957 A379768 A042469

Adjacent sequences: A393414 A393415 A393416 * A393418 A393419 A393420

Keyword

nonn,hard,more

Author

Michel Lagneau, Feb 14 2026

Extensions

a(9)-a(10) from Michel Marcus, Feb 15 2026

a(11)-a(12) from Chai Wah Wu, Mar 02 2026

Status

approved