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%I #47 Apr 25 2024 11:30:00
%S 4,8,9,22,32,33,44,55,64,77,88,93,99,422,633,775,844,933,993,4222,
%T 4442,6333,6655,6663,7533,7744,7775,8444,8884,9663,9993,44222,66333,
%U 88444,99633,99933,99993,933333,966333,996663,999993,4442222,6663333,7777775,8884444,9663333,9666633,9666663
%N 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 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 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 Michael S. Branicky, <a href="/A372034/b372034.txt">Table of n, a(n) for n = 1..197</a> (all terms with <= 21 digits)
%e The initial terms and their factorizations are:
%e 4 = [2, 2]
%e 8 = [2, 2, 2]
%e 9 = [3, 3]
%e 22 = [2, 11]
%e 32 = [2, 2, 2, 2, 2]
%e 33 = [3, 11]
%e 44 = [2, 2, 11]
%e 55 = [5, 11]
%e 64 = [2, 2, 2, 2, 2, 2]
%e 77 = [7, 11]
%e 88 = [2, 2, 2, 11]
%e 93 = [3, 31]
%e 99 = [3, 3, 11]
%e 422 = [2, 211]
%e 633 = [3, 211]
%e 775 = [5, 5, 31]
%e 844 = [2, 2, 211]
%e 933 = [3, 311]
%e 993 = [3, 331]
%e 4222 = [2, 2111]
%e 4442 = [2, 2221]
%e 6333 = [3, 2111]
%e 6655 = [5, 11, 11, 11]
%e 6663 = [3, 2221]
%e 7533 = [3, 3, 3, 3, 3, 31]
%e 7744 = [2, 2, 2, 2, 2, 2, 11, 11]
%e ...
%o (Python)
%o from sympy import factorint
%o def ni(s): return sorted(s, reverse=True) == list(s)
%o def ok(n):
%o if n < 4 or is_prime(n): return False
%o s, f = str(n), "".join(str(p)*e for p, e in factorint(n).items())
%o return ni(s+f)
%o print([k for k in range(10**6) if ok(k)]) # _Michael S. Branicky_, Apr 23 2024
%o (Python) # faster for initial segment of sequence
%o from sympy import isprime
%o from itertools import count, islice, combinations_with_replacement as mc
%o def ni(s): return s == "".join(sorted(s, reverse=True))
%o def bgen(d):
%o yield from ("".join(m) for m in mc("987654321", d))
%o def agen(): # generator of terms
%o for d in range(1, 70):
%o out = set()
%o for s in bgen(d):
%o t = int(s)
%o if t < 4 or is_prime(t): continue
%o if ni(s+"".join(str(p)*e for p, e in factorint(t).items())):
%o out.add(t)
%o yield from sorted(out)
%o print(list(islice(agen(), 50))) # _Michael S. Branicky_, Apr 23 2024
%Y Cf. A004022, A004023, A009996, A028867, A372053, A372029, A027746, A372055.
%K nonn,base
%O 1,1
%A _Scott R. Shannon_, Apr 16 2024.