A372034 For a positive number k, let L(k) denote the list consisting of k followed by the prime factors of k, with repetition, in nondecreasing order; sequence gives composite k such that the digits of L(k) are in nonincreasing order.

4, 8, 9, 22, 32, 33, 44, 55, 64, 77, 88, 93, 99, 422, 633, 775, 844, 933, 993, 4222, 4442, 6333, 6655, 6663, 7533, 7744, 7775, 8444, 8884, 9663, 9993, 44222, 66333, 88444, 99633, 99933, 99993, 933333, 966333, 996663, 999993, 4442222, 6663333, 7777775, 8884444, 9663333, 9666633, 9666663

Offset: 1

Author

Scott R. Shannon, Apr 16 2024

Keywords

• nonn
• base

Comments

Is it true that no terms end with 1? A separate search on those shows none with < 70 digits. Michael S. Branicky, Apr 23 2024

Testing all products of repunit primes (A004022, A004023), there are no terms ending in 1 less than 10^3000. - Michael S. Branicky, Apr 24 2024

Examples

			The initial terms and their factorizations are:
4 = [2, 2]
8 = [2, 2, 2]
9 = [3, 3]
22 = [2, 11]
32 = [2, 2, 2, 2, 2]
33 = [3, 11]
44 = [2, 2, 11]
55 = [5, 11]
64 = [2, 2, 2, 2, 2, 2]
77 = [7, 11]
88 = [2, 2, 2, 11]
93 = [3, 31]
99 = [3, 3, 11]
422 = [2, 211]
633 = [3, 211]
775 = [5, 5, 31]
844 = [2, 2, 211]
933 = [3, 311]
993 = [3, 331]
4222 = [2, 2111]
4442 = [2, 2221]
6333 = [3, 2111]
6655 = [5, 11, 11, 11]
6663 = [3, 2221]
7533 = [3, 3, 3, 3, 3, 31]
7744 = [2, 2, 2, 2, 2, 2, 11, 11]
...
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..197 (all terms with <= 21 digits)

Crossrefs

Cf. A004022, A004023, A009996, A028867, A372053, A372029, A027746, A372055.

Programs

Python

from sympy import factorint, isprime
def ni(s): return sorted(s, reverse=True) == list(s)
def ok(n):
    if n < 4 or isprime(n): return False
    s, f = str(n), "".join(str(p)*e for p, e in factorint(n).items())
    return ni(s+f)
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Apr 23 2024

Python

# faster for initial segment of sequence
from sympy import factorint, isprime
from itertools import islice, combinations_with_replacement as mc
def ni(s): return s == "".join(sorted(s, reverse=True))
def bgen(d):
    yield from ("".join(m) for m in mc("987654321", d))
def agen(): # generator of terms
    for d in range(1, 70):
        out = set()
        for s in bgen(d):
            t = int(s)
            if t < 4 or isprime(t): continue
            if ni(s+"".join(str(p)*e for p, e in factorint(t).items())):
                out.add(t)
        yield from sorted(out)
print(list(islice(agen(), 50))) # Michael S. Branicky, Apr 23 2024