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

1, 12, 124, 135, 1525, 13515, 124816, 12356910, 1243162124, 1525125625, 12478141928, 12510254150, 1234689111216, 1351553159265, 1597717414885, 12356910151830, 13791121336377, 123561015253050, 124510202550100, 135152575125375, 1236103206309618, 123456101215203060, 123569101518304590

Offset: 1

Author

Jaroslav Krizek, Mar 14 2010

Keywords

• nonn
• base

Comments

From Michel Marcus, Sep 25 2022: (Start)

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

Examples

			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}.
		

Links

• Tim Peters, Table of n, a(n) for n = 1..25
• David A. Corneth and Michel Marcus, Some terms found in search for terms for A357692
• David A. Corneth and Michel Marcus, Some terms <= 10^500

Crossrefs

Cf. A037278, A357692. Subsequence of A131835.

Cf. A077352, A077353.

Programs

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
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Sep 22 2022

Extensions

a(9)-a(10) from Michel Marcus, Sep 22 2022

a(11)-a(12) from Michel Marcus, Oct 02 2022

a(13)-a(15) from Tim Peters, Oct 17 2022

a(16)-a(17) from Giovanni Resta, Oct 20 2022

a(18)-a(20) from Tim Peters, Oct 27 2022

a(21) from Tim Peters, Oct 30 2022

a(22)-a(23) from Tim Peters, Nov 04 2022