A393035 Indices k such that P(k)^2 + P(k+1)^2 is prime, where P(n) is the Padovan sequence A134816(n+1).

0, 1, 2, 4, 6, 13, 16, 18, 19, 30, 40, 46, 67, 95, 102, 216, 219, 307, 580, 613, 726, 865, 3042, 8041, 8629, 8791, 8801, 12833, 17250, 23055, 58771, 74295, 117723, 144535

Offset

1,3

Comments

This sequence represents the indices k of the Padovan numbers whose sum of adjacent squares results in a prime. The primality of terms up to a(15) (index 102) was verified deterministically using SymPy's "isprime". The primality of a(23) (index 3042), a 744 digit number, was rigorously verified using the APRCL algorithm in PARI/GP. Terms from a(24) onwards are strong probable primes.

Links

Table of n, a(n) for n=1..34.

Wikipedia, Padovan sequence

Wikipedia, APRCL primality test

Example

For k = 0: P(0) = 1, P(1) = 1. 1^2 + 1^2 = 2, which is prime. So a(1) = 0.
For k = 1: P(1) = 1, P(2) = 1. 1^2 + 1^2 = 2, which is prime. So a(2) = 1.
For k = 2: P(2) = 1, P(3) = 2. 1^2 + 2^2 = 5, which is prime. So a(3) = 2.
For k = 3: P(3) = 2, P(4) = 2. 2^2 + 2^2 = 8, which is not prime.
For k = 4: P(4) = 2, P(5) = 3. 2^2 + 3^2 = 13, which is prime. So a(4) = 4

Prog

(Python)
from sympy import isprime
def a134816(n):
    # Padovan sequence generator
    a, b, c = 1, 1, 1
    for _ in range(n):
        a, b, c = b, c, a + b
    return a
def generate_sequence(limit):
    indices = []
    for k in range(limit):
        candidate = a134816(k)**2 + a134816(k+1)**2
        if isprime(candidate):
            indices.append(k)
    return indices
# The first 29 terms are found by calling generate_sequence(17251)

Crossrefs

Cf. A000040, A134816.

Sequence in context: A110980 A376644 A376533 * A058598 A332241 A319110

Adjacent sequences: A393032 A393033 A393034 * A393036 A393037 A393038

Keyword

nonn,hard,more

Author

Fırat Karayıldız, Feb 28 2026

Extensions

a(30)-a(34) from Michael S. Branicky, Mar 01 2026

Status

approved