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A346662
Number of n-digit left- or right-truncatable primes with no consecutive zero digits.
1
4, 16, 76, 300, 955, 2648, 6402, 14339, 28684, 53450, 91284, 147064, 221301, 319067, 433227, 567565, 700765, 834464, 947055, 1050886, 1114368, 1157526, 1150645, 1117265, 1044757, 963722, 855804, 753172, 633786, 528122, 426328, 339866, 264078, 202013, 150330, 111055, 78996, 56123, 38874, 26644, 17944, 11898, 7878, 4945, 3255, 2024, 1323, 764, 464, 286, 158, 77, 40, 26, 14, 5, 5, 4, 1, 1
OFFSET
1,1
COMMENTS
A left- or right-truncatable prime is a prime number from which one digit at a time may be removed from the left or right end until a single-digit prime is reached, with each digit removal resulting in a prime. There exists only one such 60-digit prime: 202075909708030901050930450609080660821035604908735717137397. Since it cannot be extended, there are no such primes with more than 60 digits, so a(60)=1 is the final term of the sequence.
EXAMPLE
The 16 two-digit left- or right-truncatable primes with no consecutive zero digits are 13, 17, 23, 29, 31, 37, 43, 47, 53, 59, 67, 71, 73, 79, 83, 97.
The first 10 three-digit left- or right-truncatable primes with no consecutive zero digits are 103, 107, 113, 131, 137, 139, 167, 173, 179, 197.
The unique 60-digit left- or right-truncatable prime with no consecutive zero digits can be sequentially truncated to a single-digit prime as follows, where each "..." indicates repeated removal of the leftmost digit:
202075909708030901050930450609080660821035604908735717137397
...
2075909708030901050930450609080660821035604908735717137397
207590970803090105093045060908066082103560490873571713739
...
970803090105093045060908066082103560490873571713739
97080309010509304506090806608210356049087357171373
...
6090806608210356049087357171373
609080660821035604908735717137
...
80660821035604908735717137
8066082103560490873571713
806608210356049087357171
...
8210356049087357171
821035604908735717
21035604908735717
2103560490873571
...
71
7
PROG
(Python)
from sympy import isprime
dumps = set({})
route = set({})
nums = [i*(10**j) for i in range(1, 10) for j in range(2)]
def addnum(a):
global route
for j in nums:
b = int("{}{}".format(a, j))
if isprime(b):
if b not in route:
route.add(b)
addnum(b)
for j in nums:
b = int("{}{}".format(j, a))
if isprime(b):
if b not in route:
route.add(b)
addnum(b)
def run():
for i in nums:
if isprime(i):
addnum(i)
run()
CROSSREFS
Left- or right-truncatable primes, excluding all 0s: A137812.
Left- or right-truncatable primes with 0s allowed, but none consecutive: A347864.
Sequence in context: A324171 A204772 A050540 * A094559 A199214 A374566
KEYWORD
nonn,fini,full
AUTHOR
Timothy Smith, Jan 25 2022
STATUS
approved