A372280 Composite numbers k such that the digits of k are in nondecreasing order while the digits of the concatenation of k's ascending order prime factors, with repetition, are in nonincreasing order.

4, 8, 9, 16, 22, 25, 27, 33, 44, 49, 55, 77, 88, 99, 125, 128, 155, 256, 279, 1477, 1555, 1688, 1899, 2799, 3479, 3577, 14777, 16888, 18999, 22599, 36799, 444577, 455777, 1112447, 1555555, 2555555, 2799999, 3577777, 3799999, 45577777, 124556677, 155555555555, 279999999999

Offset: 1

Author

Scott R. Shannon, Apr 25 2024

Keywords

• nonn
• base

Comments

A number 155...555 will be a term if it has two prime factors 5 and 3111...111. Therefore 155555555555 and 1555555555555 are both terms. See A056704.

The next term is greater than 10^11.

Examples

			444577 is a term as 444577 = 7 * 7 * 43 * 211, and 444577 has nondecreasing digits while its prime factor concatenation "7743211" has nonincreasing digits.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..64 (all terms <= 20 digits)

Crossrefs

Cf. A372308, A372034, A056704, A372029, A372055, A027746, A372249.

Programs

Python

from sympy import factorint, isprime
from itertools import count, 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("0123456789", d) if m[0]!="0")
def agen(): # generator of terms
    for d in count(1):
        for s in bgen(d):
            t = int(s)
            if t < 4 or isprime(t): continue
            if ni("".join(str(p)*e for p,e in factorint(t).items())):
                yield t
print(list(islice(agen(), 41))) # Michael S. Branicky, Apr 26 2024

Extensions

a(42)-a(43) from Michael S. Branicky, Apr 26 2024