A372308 Composite numbers k such that the digits of k are in nonincreasing order while the digits of the concatenation of k's ascending order prime factors, with repetition, are in nondecreasing order.

4, 6, 8, 9, 10, 20, 21, 30, 32, 40, 42, 50, 54, 60, 63, 64, 70, 72, 74, 75, 80, 81, 84, 90, 92, 94, 96, 98, 100, 111, 200, 210, 222, 300, 320, 333, 400, 420, 432, 441, 444, 500, 531, 540, 553, 554, 600, 611, 630, 632, 640, 666, 700, 711, 720, 750, 752, 800, 810, 840, 851, 864, 871, 875, 882

Offset: 1

Author

Scott R. Shannon, Apr 26 2024

Keywords

• nonn
• base

Comments

As all the numbers 10,20,...,90,100 are terms, all numbers that are recursively 10 times these values are also terms as they just add an additional 2 and 5 to their parent's prime factor list.

A number 999...9998 will be a term if it has two prime factors 2 and 4999...999. Therefore 999999999999998 and 999...9998 (with 54 9's) are both terms. See A056712.

Examples

			42 is a term as 42 = 2 * 3 * 7, and 42 has nonincreasing digits while its prime factor concatenation "237" has nondecreasing digits.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..884 (terms 1..458 from Scott R. Shannon; all terms < 10^18)

Crossrefs

Cf. A372295, A372280, A056712, A372034, A372029, A372055, A027746, A372249.

Programs

Python

from sympy import factorint, isprime
from itertools import count, islice, combinations_with_replacement as mc
def nd(s): return s == "".join(sorted(s))
def bgen(d):
    yield from ("".join(m) for m in mc("9876543210", d) if m[0]!="0")
def agen(): # generator of terms
    for d in count(1):
        out = set()
        for s in bgen(d):
            t = int(s)
            if t < 4 or isprime(t): continue
            if nd("".join(str(p)*e for p,e in factorint(t).items())):
                out.add(t)
        yield from sorted(out)
print(list(islice(agen(), 65))) # Michael S. Branicky, Apr 26 2024