A383779 Primes where successively deleting the least significant digit yields a sequence that alternates between a prime and a nonprime at every step until a single-digit number remains.

2, 3, 5, 7, 11, 13, 17, 19, 41, 43, 47, 61, 67, 83, 89, 97, 211, 223, 227, 229, 241, 251, 257, 263, 269, 271, 277, 281, 283, 307, 331, 337, 347, 349, 353, 359, 367, 383, 389, 397, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 701, 709, 727, 743, 751, 757, 761, 769, 773, 787

Offset: 1

Author

Stefano Spezia, May 09 2025

Keywords

• nonn
• base
• fini

Comments

Is this sequence infinite?

Likely no, see A383780 and comment there. - Michael S. Branicky, May 11 2025

From Michael S. Branicky, May 16 2025: (Start)

Sequence is finite with 3356513448 terms (cf. A383780).

Last term: 70123916363515199416199518301698321195339012727994799190371992151279729974757397909992327936943877127375781091143 (End)

Examples

			211 is a term since 211 is a prime, 21 is a nonprime, and 2 is a prime;
23 is not a term since 23 and 2 are both prime.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..20000
• Michael S. Branicky, Largest even-length (= 108 digits) then odd-length (= 113 digits) terms

Crossrefs

Cf. A024770, A069090, A383780, A383781.

Programs

Mathematica

Unprotect[CompositeQ]; CompositeQ[1]:=True; Protect[CompositeQ]; Q[n_]:=And[AllTrue[FromDigits/@Table[Take[IntegerDigits[n], i], {i,IntegerLength[n],1,-2}], PrimeQ], AllTrue[FromDigits/@Table[Take[IntegerDigits[n], i], {i,IntegerLength[n]-1,1,-2}],CompositeQ]]; Select[Prime[Range[140]], Q]

Python

from gmpy2 import is_prime, mpz
from itertools import count, islice
def agen():
    olst, elst = [2, 3, 5, 7], [11, 13, 17, 19, 41, 43, 47, 61, 67, 83, 89, 97]
    yield from olst + elst
    for n in count(1):
        olst2, elst2 = [], []
        for o in olst:
            for i in range(1, 100, 2):
                t = 100*o + i
                if is_prime(t) and not is_prime(t//10):
                    olst2.append(t)
        yield from olst2
        for e in elst:
            for i in range(100):
                t = 100*e + i
                if is_prime(t) and not is_prime(t//10):
                    elst2.append(t)
        yield from elst2
        olst, elst = olst2, elst2
print(list(islice(agen(), 70))) # Michael S. Branicky, May 11 2025

Python

# predicate test useful for large n (cf. a-file of largest terms)
from gmpy2 import digits, is_prime, mpz
def ok(n):
    s = digits(n)
    return is_prime(n) and all(int(is_prime(mpz(s[:-i]))) == 1-i&1 for i in range(1, len(s)))
    # Michael S. Branicky, May 16 2025