A362417 Beginning with 1, smallest positive integer not yet in the sequence such that two adjacent digits A and B of the sequence (also ignoring commas between terms) produce a prime = A + 2B. This is the earliest infinitely extensible such sequence.

1, 3, 5, 7, 20, 11, 9, 13, 15, 19, 50, 111, 30, 113, 51, 31, 35, 37, 53, 57, 59, 70, 115, 73, 75, 91, 95, 97, 201, 119, 120, 130, 131, 135, 137, 301, 150, 151, 153, 157, 311, 159, 191, 195, 197, 313, 501, 315, 319, 511, 320, 1111, 350, 1113, 513, 515, 351, 353

Offset

1,2

Comments

The integer 10 is the first one that will never appear in the sequence (as the result of 1 + 2*0 is not a prime). The next absent will be 14.

From Michael S. Branicky, Apr 19 2023: (Start)

The only allowed pairs of digits AB are 01, 11, 12, 13, 15, 16, 18, 19, 20, 30, 31, 32, 34, 35, 37, 38, 50, 51, 53, 54, 56, 57, 59, 70, 72, 73, 75, 76, 78, 91, 92, 94, 95, 97.

Further, any appearance of 4, 6, or 8 as a digit would end the sequence, as would a term with last digit 2 (since the next term cannot start with 0).

As long as no term ends in 2, 4, 6, or 8, the sequence is infinitely extensible since the edge and cycle 01 -> 13 -> 31 -> 13 (at least) can be used to extend terms ending in 0, 1, or 3; and 75 -> 59 -> 97 to extend terms ending in 5, 7, or 9. (End)

Links

Table of n, a(n) for n=1..58.

Example

Digit A = 1 and B = 3 lead to 7 (prime) = A+2B;
Digit A = 3 and B = 5 lead to 13 (prime) = A+2B;
Digit A = 5 and B = 7 lead to 19 (prime) = A+2B;
Digit A = 7 and B = 2 lead to 11 (prime) = A+2B;
Digit A = 2 and B = 0 lead to 2 (prime) = A+2B;
Digit A = 0 and B = 1 lead to 2 (prime) = A+2B;
Digit A = 1 and B = 1 lead to 3 (prime) = A+2B; etc.

Prog

(Python)
from sympy import isprime
from itertools import islice
def c(s):
    if s[-1] == "2" or "4" in s or "6" in s or "8" in s: return False
    return all(isprime(int(s[i])+2*int(s[i+1])) for i in range(len(s)-1))
def agen(): # generator of terms
    last, aset = "1", {1}
    yield 1
    while True:
        k = 2
        while k in aset or not c(last+str(k)): k += 1
        an = k; yield an; last += str(an); aset.add(an)
print(list(islice(agen(), 58))) # Michael S. Branicky, Apr 19 2023

Crossrefs

Cf. A182178 (B is multiplied by 1), A362418 (B is multiplied by 3).

Sequence in context: A171643 A072002 A217651 * A352710 A163637 A336985

Adjacent sequences: A362414 A362415 A362416 * A362418 A362419 A362420

Keyword

base,nonn

Author

Eric Angelini, Apr 19 2023

Extensions

a(7) inserted and a(30) and beyond from Michael S. Branicky, Apr 19 2023

Status

approved