A383780 a(n) is the number of n-digit terms in A383779.

4, 12, 46, 103, 396, 717, 2451, 3929, 11803, 17202, 46916, 62668, 157138, 197114, 458064, 541267, 1180018, 1323543, 2718398, 2915696, 5675113, 5839596, 10821575, 10724938, 18983655, 18174231, 30856021, 28608908, 46708476, 42036009, 66157433, 57908390, 88020231, 75070514

Offset: 1

Author

Stefano Spezia, May 09 2025

Keywords

• nonn
• base

Comments

Conjecture: A383779 is finite, since after the first two terms here, the ratios between successive odd terms and the ratios between successive even terms are quickly decreasing. - Michael S. Branicky, May 11 2025

Indeed, a(n) < a(n-2) for n = 45..55 (see b-file). - Michael S. Branicky, May 15 2025

A383779 is finite with 3356513448 terms (summing terms of b-file); a(n) = 0 for n >= 114. - Michael S. Branicky, May 16 2025

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..115

Crossrefs

Cf. A383779.

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]]; a[n_]:=Module[{p=Prime[PrimePi[10^(n-1)]+1], k=0}, While[10^(n-1)<=p<10^n-1, If[Q[p], k++]; p=NextPrime[p]]; k]; Array[a,7]

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 (len(olst), len(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 len(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 len(elst2)
        olst, elst = olst2, elst2
print(list(islice(agen(), 16))) # Michael S. Branicky, May 11 2025

Extensions

a(12) and beyond from Michael S. Branicky, May 11 2025