A394059 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) = prime(k) mod 6 = A039704(k).

1, 9, 3, 8, 3, 30, 3, 29, 16, 87, 23, 86, 22, 99, 140, 98, 140, 717, 140, 716, 140, 715, 139, 714, 3934, 12862, 3933, 26993, 5778, 26992, 20246, 348411, 20245, 488970, 20244, 488969, 20243, 488968, 76776, 1822976, 76775, 1822975, 76774, 3540352, 76773, 10884806, 5673605, 19624687, 5673604

Offset

1,2

Comments

Conjecture: There exist infinitely many palindromes of every length in the sequence of residues of primes modulo 6.

The aim of this sequence is to study the local symmetries of the sequence A039704. To do this, we identify palindromes in subsequences of A039704.

Links

Sean A. Irvine, Table of n, a(n) for n = 1..59

Example

Initial solutions:
  n  a(n) palindrome
  --------------------------------------------
  1   1  [1]
  2   9  [5, 5]
  3   3  [5, 1, 5]
  4   8  [1, 5, 5, 1]
  5   3  [5, 1, 5, 1, 5]
  6  30  [5, 1, 5, 5, 1, 5]
  7   3  [5, 1, 5, 1, 5, 1, 5]
  8  29  [1, 5, 1, 5, 5, 1, 5, 1]
  9  16  [5, 5, 1, 1, 5, 1, 1, 5, 5]
 10  87  [5, 1, 5, 1, 5, 5, 1, 5, 1, 5]
 11  23  [5, 5, 1, 5, 1, 5, 1, 5, 1, 5, 5]
 12  86  [5, 5, 1, 5, 1, 5, 5, 1, 5, 1, 5, 5]

Maple

with(numtheory):
Nmax := 50: Pmax := 500000:
b := Array(1..Pmax):
for k from 1 to Pmax do b[k] := irem(ithprime(k), 6) end do:
a := Array(1..Nmax, fill=infinity):
for k from 1 to Pmax do
r := 0:
while k-r >=1 and k+r <= Pmax and b[k-r]=b[k+r] do
n := 2*r+1: start := k-r:
if n <= Nmax and start < a[n] then a[n] := start: end if:
r := r+1:
end do:
r := 0:
while k-r >=1 and k+r+1 <= Pmax and b[k-r]=b[k+r+1] do
n := 2*r+2: start := k-r:
if n <= Nmax and start < a[n] then a[n] := start: end if:
r := r+1:
end do:
end do:
seq(a[n], n=1..Nmax);

Prog

(PARI) a(n, v) = for (i=1, #v-n, my(w = vector(n, k, v[i+k-1])); if (Vecrev(w) == w, return(i)); );
lista(nn) = my(v = primes(nn) % 6); for (n=1, 50, my(x = a(n, v)); if (x, print1(x, ", "), return));
lista(10^7) \\ Michel Marcus, Mar 10 2026

Crossrefs

Cf. A039704, A393703.

Sequence in context: A394833 A284832 A380599 * A387979 A358942 A266559

Adjacent sequences: A394056 A394057 A394058 * A394060 A394061 A394062

Keyword

nonn

Author

Michel Lagneau, Mar 09 2026

Extensions

More terms from Michel Marcus, Mar 10 2026

Status

approved