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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.
7
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
1,1
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.
LINKS
Michael S. Branicky, Table of n, a(n) for n = 1..64 (all terms <= 20 digits)
EXAMPLE
444577 is a term as 444577 = 7 * 7 * 43 * 211, and 444577 has nondecreasing digits while its prime factor concatenation "7743211" has nonincreasing digits.
PROG
(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
KEYWORD
nonn,base
AUTHOR
Scott R. Shannon, Apr 25 2024
EXTENSIONS
a(42)-a(43) from Michael S. Branicky, Apr 26 2024
STATUS
approved