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

1,1

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.

Links

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

Example

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

Prog

(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

Crossrefs

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

Sequence in context: A202265 A004716 A212372 * A320726 A205530 A100373

Adjacent sequences: A372305 A372306 A372307 * A372309 A372310 A372311

Keyword

nonn,base

Author

Scott R. Shannon, Apr 26 2024

Status

approved