|
|
A175252
|
|
Numbers whose digit representation is equal to the digit representation of the initial terms of their sets of divisors in increasing order.
|
|
7
|
|
|
1, 12, 124, 135, 1525, 13515, 124816, 12356910, 1243162124, 1525125625, 12478141928, 12510254150, 1234689111216, 1351553159265, 1597717414885, 12356910151830, 13791121336377, 123561015253050, 124510202550100, 135152575125375, 1236103206309618, 123456101215203060, 123569101518304590
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
The term 124 (2^2*31) corresponds to the term of A077352 that is a prime.
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)
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
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
a(26) > 10^23 (there are no terms with 23 digits). - Tim Peters, Dec 21 2022
|
|
LINKS
|
|
|
EXAMPLE
|
a(1) = 1: d(1) = {1}.
a(2) = 12: d(12) = {1, 2, 3, 4, 6, 12}.
a(3) = 124: d(124) = {1, 2, 4, 31, 62, 124}.
a(4) = 135: d(135) = {1, 3, 5, 9, 15, 27, 45, 135}.
|
|
PROG
|
(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
(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
(Python)
from sympy import divisors
def ok(n):
target, s = str(n), ""
if target[0] != "1": return False
for d in divisors(n):
s += str(d)
if len(s) >= len(target): return s == target
elif not target.startswith(s): return False
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,base
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|