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

1,2

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

Links

Table of n, a(n) for n=1..25.

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A037278, A175252 (a subsequence).

Cf. A004022 (a subsequence).

Subsequence of A131835.

Sequence in context: A108218 A038326 A249312 * A041059 A041260 A109665

Adjacent sequences: A357270 A357271 A357272 * A357274 A357275 A357276

Keyword

nonn,base

Author

Michel Marcus, Sep 22 2022

Extensions

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

Status

approved