A343834 Primes with digits in nondecreasing order, only primes, and with sum of digits also a prime.

2, 3, 5, 7, 23, 223, 227, 337, 557, 577, 2333, 2357, 2377, 2557, 2777, 33377, 222337, 222557, 233357, 233777, 235577, 2222333, 2233337, 2235557, 3337777, 3355777, 5555777, 22222223, 22233577, 23333357, 23377777, 25577777, 222222227, 222222557, 222222577

Offset: 1

Author

Mikk Heidemaa, May 01 2021

Keywords

• nonn
• base
• easy

Comments

Intersection of A028864 and A062088.

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A019546, A028864, A046704, A062088.

Programs

Mathematica

a[p_] := With[{dg = IntegerDigits@p}, PrimeQ@p && OrderedQ@dg && AllTrue[dg, PrimeQ] && PrimeQ@ Total@dg]; Cases[ Range[3*10^7], _?(a@# &)] (* or *)
upToDigitLen[k_] := Cases[ FromDigits@# & /@ Select[ Flatten[ Table[ Tuples[{2, 3, 5, 7}, {i}], {i, k}], 1], OrderedQ[#] &], _?(PrimeQ@# && PrimeQ@ Total@ IntegerDigits@# &)]; upToDigitLen[10]

Python

from sympy import isprime
from sympy.utilities.iterables import multiset_combinations
def aupton(terms):
  n, digits, alst = 0, 1, []
  while len(alst) < terms:
    mcstr = "".join(d*digits for d in "2357")
    for mc in multiset_combinations(mcstr, digits):
      sd = sum(int(d) for d in mc)
      if not isprime(sd): continue
      t = int("".join(mc))
      if isprime(t): alst.append(t)
      if len(alst) == terms: break
    else: digits += 1
  return alst
print(aupton(35)) # Michael S. Branicky, May 01 2021

Extensions

a(33) and beyond from Michael S. Branicky, May 01 2021