A357273 Integers m whose decimal expansion is a prefix of the concatenation of the divisors of m.

1, 11, 12, 124, 135, 1111, 1525, 13515, 124816, 1223462, 12356910, 13919571, 1210320658, 1243162124, 1525125625, 12346121028, 12478141928, 12510153130, 12510254150, 1234689111216, 1351553159265, 1597717414885, 1713913539247, 12356910151830, 13791121336377

Offset: 1

Author

Michel Marcus, Sep 22 2022

Keywords

• nonn
• base

Comments

This is different from A175252 in that the digits to be concatenated can end in the middle of a divisor.

All terms start with the digit 1. - Chai Wah Wu, Sep 23 2022

a(26) > 10^14. - Giovanni Resta, Oct 20 2022

Examples

			11 is a term since its divisors are 1 and 11 whose concatenation is 111 whose first 2 digits are 11.
1111 is a term since its divisors are 1, 11, 101, and 1111 whose concatenation is 1111011111 whose first 4 digits are 1111.
		

Crossrefs

Cf. A037278, A175252 (a subsequence).

Cf. A004022 (a subsequence).

Subsequence of A131835.

Programs

Mathematica

q[n_] := (Join @@ IntegerDigits @ Divisors[n])[[1 ;; Length @ (d = IntegerDigits[n])]] == d; Select[Range[1.3*10^6], q] (* Amiram Eldar, Sep 22 2022 *)

PARI

f(n) = my(s=""); fordiv(n, d, s = concat(s, Str(d))); s; \\ A037278
isok(k) = if (k==1, 1, my(v=strsplit(f(k), Str(k))); (v[1] == ""));

Python

from sympy import divisors
def ok(n): return "".join(str(d) for d in divisors(n)).startswith(str(n))
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Sep 22 2022

Python

from itertools import count, islice
from sympy import divisors
def A357273_gen(): # generator of terms
    yield 1
    for n in count(1):
        r = str(n)
        if n&1 or r.startswith('2'):
            m = 10**(c:=len(r))+n
            s, sm = '', str(m)
            for d in divisors(m):
                s += str(d)
                if len(s) >= c+1:
                    break
            if s.startswith(sm):
                yield m
A357273_list = list(islice(A357273_gen(),10)) # Chai Wah Wu, Sep 23 2022

Extensions

a(16)-a(25) from Giovanni Resta, Oct 20 2022