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A372034 For a positive number k, let L(k) denote the list consisting of k followed by the prime factors of k, with repetition, in nondecreasing order; sequence gives composite k such that the digits of L(k) are in nonincreasing order.

%I A372034 #56 Jul 09 2025 05:03:58
%S A372034 4,8,9,22,32,33,44,55,64,77,88,93,99,422,633,775,844,933,993,4222,
%T A372034 4442,6333,6655,6663,7533,7744,7775,8444,8884,9663,9993,44222,66333,
%U A372034 88444,99633,99933,99993,933333,966333,996663,999993,4442222,6663333,7777775,8884444,9663333,9666633,9666663
%N A372034 For a positive number k, let L(k) denote the list consisting of k followed by the prime factors of k, with repetition, in nondecreasing order; sequence gives composite k such that the digits of L(k) are in nonincreasing order.
%C A372034 Is it true that no terms end with 1?  A separate search on those shows none with < 70 digits. _Michael S. Branicky_,  Apr 23 2024
%C A372034 Testing all products of repunit primes (A004022, A004023), there are no terms ending in 1 less than 10^3000. - _Michael S. Branicky_, Apr 24 2024
%H A372034 Michael S. Branicky, <a href="/A372034/b372034.txt">Table of n, a(n) for n = 1..197</a> (all terms with <= 21 digits)
%e A372034 The initial terms and their factorizations are:
%e A372034 4 = [2, 2]
%e A372034 8 = [2, 2, 2]
%e A372034 9 = [3, 3]
%e A372034 22 = [2, 11]
%e A372034 32 = [2, 2, 2, 2, 2]
%e A372034 33 = [3, 11]
%e A372034 44 = [2, 2, 11]
%e A372034 55 = [5, 11]
%e A372034 64 = [2, 2, 2, 2, 2, 2]
%e A372034 77 = [7, 11]
%e A372034 88 = [2, 2, 2, 11]
%e A372034 93 = [3, 31]
%e A372034 99 = [3, 3, 11]
%e A372034 422 = [2, 211]
%e A372034 633 = [3, 211]
%e A372034 775 = [5, 5, 31]
%e A372034 844 = [2, 2, 211]
%e A372034 933 = [3, 311]
%e A372034 993 = [3, 331]
%e A372034 4222 = [2, 2111]
%e A372034 4442 = [2, 2221]
%e A372034 6333 = [3, 2111]
%e A372034 6655 = [5, 11, 11, 11]
%e A372034 6663 = [3, 2221]
%e A372034 7533 = [3, 3, 3, 3, 3, 31]
%e A372034 7744 = [2, 2, 2, 2, 2, 2, 11, 11]
%e A372034 ...
%o A372034 (Python)
%o A372034 from sympy import factorint, isprime
%o A372034 def ni(s): return sorted(s, reverse=True) == list(s)
%o A372034 def ok(n):
%o A372034     if n < 4 or isprime(n): return False
%o A372034     s, f = str(n), "".join(str(p)*e for p, e in factorint(n).items())
%o A372034     return ni(s+f)
%o A372034 print([k for k in range(10**6) if ok(k)]) # _Michael S. Branicky_, Apr 23 2024
%o A372034 (Python) # faster for initial segment of sequence
%o A372034 from sympy import factorint, isprime
%o A372034 from itertools import islice, combinations_with_replacement as mc
%o A372034 def ni(s): return s == "".join(sorted(s, reverse=True))
%o A372034 def bgen(d):
%o A372034     yield from ("".join(m) for m in mc("987654321", d))
%o A372034 def agen(): # generator of terms
%o A372034     for d in range(1, 70):
%o A372034         out = set()
%o A372034         for s in bgen(d):
%o A372034             t = int(s)
%o A372034             if t < 4 or isprime(t): continue
%o A372034             if ni(s+"".join(str(p)*e for p, e in factorint(t).items())):
%o A372034                 out.add(t)
%o A372034         yield from sorted(out)
%o A372034 print(list(islice(agen(), 50))) # _Michael S. Branicky_, Apr 23 2024
%Y A372034 Cf. A004022, A004023, A009996, A028867, A372053, A372029, A027746, A372055.
%K A372034 nonn,base
%O A372034 1,1
%A A372034 _Scott R. Shannon_, Apr 16 2024