A277997 Lexicographically first sequence of nonprimes (with no duplicates) whose absolute first differences are prime (with no duplicates).

1, 4, 6, 25, 8, 15, 10, 21, 34, 57, 14, 45, 16, 63, 22, 75, 38, 99, 20, 87, 28, 111, 40, 129, 26, 123, 50, 159, 32, 133, 240, 49, 162, 299, 18, 169, 12, 143, 282, 55, 204, 371, 24, 187, 360, 77, 256, 27, 208, 9, 202, 399, 46, 279, 56, 267, 506, 39, 280, 531, 44, 301, 30, 323, 54, 361, 48, 325, 62, 393, 76, 387, 724, 33

Offset

1,2

Comments

The sequence starts with a(1) and was always extended with the smallest integer not yet present that does not lead to a contradiction. It is conjectured that the sequence together with its first absolute differences is a permutation of the positive integers.

Links

Jean-Marc Falcoz, Table of n, a(n) for n = 1..5001

Éric Angelini, Absolute first differences are a permutation of the primes, digest of SeqFan discussion, 2010-10-01.

Éric Angelini, Absolute first differences are a permutation of the primes, local copy of SeqFan discussion, 2010-10-01.

Example

After a(1) = 1, we cannot have a(2) = 2 or a(2) = 3 as these are prime numbers; a(2) = 4 is ok as the absolute difference |1-4| = 3 is prime. The next term a(3); cannot be 5 as 5 is prime; a(3) = 6 is ok as the absolute difference |4-6| = 2 is a prime not yet present in the absolute first differences. The next term a(4) cannot be 7 as 7 is prime; nor a(4) = 8 as |6-8| = 2 is a prime already present in the absolute first differences; nor a(4) = 9 as |6-9| = 3 is a prime already present in the first absolute differences; nor a(4) = 10 as |6-10| = 4 is not a prime; nor a(4) = 11 as 11 is prime; nor a(4) = 12 as |6-12| = 4 is not a prime; nor a(4) = 13 as 13 is a prime; nor a(4) = 14 as |6-14| = 4 is not a prime; nor a(4) = 15 as |6-15| = 9 is not a prime; nor a(4) = 16 as |6-16| = 10 is not a prime; nor a(4) = 17 as 17 is a prime; nor a(4) = 18 as |6-18| = 12 is not prime; nor a(4) = 19 as 19 is a prime; nor a(4) = 20 as |6-20| = 14 is not a prime; nor a(4) = 21 as |6-21| = 15 is not a prime; nor a(4) = 22 as |6-22| = 16 is not a prime; nor a(4) = 23 as 23 is a prime; nor a(4) = 24 as |6-24| = 18 is not a prime; a(4) = 25 is ok as |6-25| = 19 is a prime not yet present in the absolute first differences, etc.

Prog

(Python)
from sympy import isprime
from itertools import count, islice
def nextcomposite(n): return next(k for k in count(n+1) if not isprime(k))
def agen(): # generator of terms
    an, a, nextc = 1, {1}, 4
    while True:
        yield an
        c = nextc
        while c in a or not isprime(abs(c-an)) or abs(c-an) in a:
            c = nextcomposite(c)
        a.add(c)
        a.add(abs(c-an))
        an = c
        while nextc in a: nextc = nextcomposite(nextc)
print(list(islice(agen(), 74))) # Michael S. Branicky, Dec 23 2024

Crossrefs

Sequence in context: A165164 A241602 A136591 * A009459 A239625 A123055

Adjacent sequences: A277994 A277995 A277996 * A277998 A277999 A278000

Keyword

nonn

Author

Eric Angelini and Jean-Marc Falcoz, Nov 08 2016

Status

approved