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A175252 Numbers whose digit representation is equal to the digit representation of the initial terms of their sets of divisors in increasing order.

%I A175252 #150 Dec 21 2022 20:41:08
%S A175252 1,12,124,135,1525,13515,124816,12356910,1243162124,1525125625,
%T A175252 12478141928,12510254150,1234689111216,1351553159265,1597717414885,
%U A175252 12356910151830,13791121336377,123561015253050,124510202550100,135152575125375,1236103206309618,123456101215203060,123569101518304590
%N A175252 Numbers whose digit representation is equal to the digit representation of the initial terms of their sets of divisors in increasing order.
%C A175252 From _Michel Marcus_, Sep 25 2022: (Start)
%C A175252 The term 124 (2^2*31) corresponds to the term of A077352 that is a prime.
%C A175252 The terms 135 (5*3^3), 1525 (5^2*61) and 1525125625 (5^4*2440201) correspond to the terms of A077353 that are powers of primes. (End)
%C A175252 The term 1597717414885 = 5 * 977 * 1741 * 187861, found by _David A. Corneth_, is especially remarkable for the magnitude of its 2nd smallest prime factor (counting repetitions). - _Peter Munn_, Oct 10 2022
%C A175252 Define g(n) to be the LCM of the divisors of a(n) that appear in the digit string of a(n) as specified in the definition, and let f(n) = log(g(n))/log(a(n)). Are there are only finitely many n for which f(n) >= f(4) = log(15)/log(135) = 0.55206901...? - _Peter Munn_, Oct 19 2022
%C A175252 a(26) > 10^23 (there are no terms with 23 digits). - _Tim Peters_, Dec 21 2022
%H A175252 Tim Peters, <a href="/A175252/b175252.txt">Table of n, a(n) for n = 1..25</a>
%H A175252 David A. Corneth and Michel Marcus, <a href="/A175252/a175252_5.gp.txt">Some terms found in search for terms for A357692</a>
%H A175252 David A. Corneth and Michel Marcus, <a href="/A175252/a175252_6.gp.txt">Some terms <= 10^500</a>
%e A175252 a(1) = 1: d(1) = {1}.
%e A175252 a(2) = 12: d(12) = {1, 2, 3, 4, 6, 12}.
%e A175252 a(3) = 124: d(124) = {1, 2, 4, 31, 62, 124}.
%e A175252 a(4) = 135: d(135) = {1, 3, 5, 9, 15, 27, 45, 135}.
%o A175252 (PARI) isok(k) = my(s=""); fordiv(k, d, s=concat(s, Str(d)); if (eval(s)==k, return(1)); if (eval(s)> k, return(0))); \\ _Michel Marcus_, Sep 22 2022
%o A175252 (PARI) is(n, {u = 10^5}) = { my(oldu = u, s, d, fe); s = ""; u = min(u, n); fe = factor(n, u); d = divisors(fe); if(#fe~ > 0 && fe[#fe~, 1] > u, d = select(x -> x < fe[#fe~, 1], d); ); for(i = 1, #d, if(d[i] > u, return(is(n, 10*oldu)); ); s=concat(s, Str(d[i])); if(eval(s) == n, return(1)); if(eval(s) > n, return(0)); ); is(n, 10*oldu); } \\ _David A. Corneth_, Oct 12 2022, Nov 07 2022
%o A175252 (Python)
%o A175252 from sympy import divisors
%o A175252 def ok(n):
%o A175252     target, s = str(n), ""
%o A175252     if target[0] != "1": return False
%o A175252     for d in divisors(n):
%o A175252         s += str(d)
%o A175252         if len(s) >= len(target): return s == target
%o A175252         elif not target.startswith(s): return False
%o A175252 print([k for k in range(10**6) if ok(k)]) # _Michael S. Branicky_, Sep 22 2022
%Y A175252 Cf. A037278, A357692. Subsequence of A131835.
%Y A175252 Cf. A077352, A077353.
%K A175252 nonn,base
%O A175252 1,2
%A A175252 _Jaroslav Krizek_, Mar 14 2010
%E A175252 a(9)-a(10) from _Michel Marcus_, Sep 22 2022
%E A175252 a(11)-a(12) from _Michel Marcus_, Oct 02 2022
%E A175252 a(13)-a(15) from _Tim Peters_, Oct 17 2022
%E A175252 a(16)-a(17) from _Giovanni Resta_, Oct 20 2022
%E A175252 a(18)-a(20) from _Tim Peters_, Oct 27 2022
%E A175252 a(21) from _Tim Peters_, Oct 30 2022
%E A175252 a(22)-a(23) from _Tim Peters_, Nov 04 2022