A362418 -id:A362418 - cp's OEIS frontend

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

Eric Angelini, Apr 19 2023

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)

			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.

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

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

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

cp's OEIS Frontend

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.

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