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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.

%I A372308 #25 Apr 27 2024 03:37:32
%S A372308 4,6,8,9,10,20,21,30,32,40,42,50,54,60,63,64,70,72,74,75,80,81,84,90,
%T A372308 92,94,96,98,100,111,200,210,222,300,320,333,400,420,432,441,444,500,
%U A372308 531,540,553,554,600,611,630,632,640,666,700,711,720,750,752,800,810,840,851,864,871,875,882
%N 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.
%C A372308 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.
%C A372308 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.
%H A372308 Michael S. Branicky, <a href="/A372308/b372308.txt">Table of n, a(n) for n = 1..884</a> (terms 1..458 from Scott R. Shannon; all terms < 10^18)
%e A372308 42 is a term as 42 = 2 * 3 * 7, and 42 has nonincreasing digits while its prime factor concatenation "237" has nondecreasing digits.
%o A372308 (Python)
%o A372308 from sympy import factorint, isprime
%o A372308 from itertools import count, islice, combinations_with_replacement as mc
%o A372308 def nd(s): return s == "".join(sorted(s))
%o A372308 def bgen(d):
%o A372308     yield from ("".join(m) for m in mc("9876543210", d) if m[0]!="0")
%o A372308 def agen(): # generator of terms
%o A372308     for d in count(1):
%o A372308         out = set()
%o A372308         for s in bgen(d):
%o A372308             t = int(s)
%o A372308             if t < 4 or isprime(t): continue
%o A372308             if nd("".join(str(p)*e for p,e in factorint(t).items())):
%o A372308                 out.add(t)
%o A372308         yield from sorted(out)
%o A372308 print(list(islice(agen(), 65))) # _Michael S. Branicky_, Apr 26 2024
%Y A372308 Cf. A372295, A372280, A056712, A372034, A372029, A372055, A027746, A372249.
%K A372308 nonn,base
%O A372308 1,1
%A A372308 _Scott R. Shannon_, Apr 26 2024