A383781 Primes where successively deleting the most 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, 19, 29, 31, 41, 59, 61, 71, 79, 89, 127, 157, 163, 193, 227, 233, 257, 263, 277, 293, 433, 457, 463, 487, 557, 563, 577, 587, 593, 677, 727, 733, 757, 787, 827, 857, 863, 877, 887, 977, 1129, 1171, 1231, 1259, 1279, 1289, 1319, 1361, 1429, 1459

Offset

1,1

Comments

Is this sequence infinite?

See A383782 and the comment therein. - Michael S. Branicky, May 11 2025

Links

Table of n, a(n) for n=1..54.

Example

127 is a term since 127 is a prime, 27 is a nonprime, and 7 is a prime;
13 is not a term since 13 and 3 are both prime.

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[240]], Q]

Prog

(Python)
from gmpy2 import is_prime, mpz
from itertools import count, islice
def agen():
    olst, elst = [2, 3, 5, 7], [11, 19, 29, 31, 41, 59, 61, 71, 79, 89]
    for n in count(1):
        yield from sorted(olst + elst)
        olst2, elst2 = [], []
        for o in olst:
            o, base = o, 10**(2*n-1)
            for i in range(10*base, 100*base, base):
                t = i + o
                t2 = int(str(t)[1:])
                if is_prime(t) and not is_prime(t2):
                    olst2.append(t)
        for e in elst:
            e, base = e, 10**(2*n)
            for i in range(10*base, 100*base, base):
                t = i + e
                t2 = int(str(t)[1:])
                if is_prime(t) and not is_prime(t2):
                    elst2.append(t)
        olst, elst = olst2, elst2
print(list(islice(agen(), 68))) # Michael S. Branicky, May 11 2025

Crossrefs

Cf. A024785, A383780, A383782.

Sequence in context: A396480 A083771 A158069 * A039726 A340418 A115617

Adjacent sequences: A383778 A383779 A383780 * A383782 A383783 A383784

Keyword

nonn,base

Author

Stefano Spezia, May 09 2025

Status

approved