A175252 -id:A175252 - cp's OEIS frontend

A357692 Integers k such that A037278(k) is a term of A175252.

1, 2, 4, 15, 16, 25, 60, 90, 100, 124, 150, 240, 375, 384, 600, 618, 625, 960, 1536, 3330, 3750, 4650, 5760, 10000, 10500, 10752, 15000, 16384, 17500, 24576, 25600, 40000, 49500, 62500, 102400, 139200, 168750, 198400, 323280, 526848, 960000, 1179648, 1248000, 1369125

Offset: 1

Michel Marcus, Oct 10 2022

			A037278(4) = 124, a term of A175252.
A037278(15) = 13515, a term of A175252.
A037278(16) = 124816, a term of A175252.
A037278(25) = 1525, a term of A175252.

David A. Corneth, Table of n, a(n) for n = 1..81

Subsequence of A069872.

Cf. A037278, A175252.

is(n, {u = 10^5}) = {my(e = eval(concat(concat([""], divisors(n))))); if(e % n != 0, return(0); ); my(oldu = u, s, d); u = min(e, u); s = ""; d = divisors(factor(e, u)); d = select(x -> x <= u, d); for(i = 1, #d, s=concat(s, Str(d[i])); if(eval(s) == e, return(1)); if(eval(s) > e, return(0)); ); is(n, 10*oldu); } \\ David A. Corneth, Oct 10 2022, adapted from Michel Marcus' isok at A175252

More terms from David A. Corneth, Oct 10 2022

A357428 Numbers whose digit representation in base 2 is equal to the digit representation in base 2 of the initial terms of their sets of divisors in increasing order.

Original entry on oeis.org

1, 6, 52, 63, 222, 2037, 6776, 26896, 124641, 220336192, 222066488

Offset: 1

Michel Marcus, Sep 28 2022

a(1), a(2), a(3), a(8) and a(10) belong to A164894; A164894(13) = 2032242676629600594233921536, A164894(19) = 1288086824419468350412109535086131006200927555108489920512 and A164894(29) are also terms. - Rémy Sigrist, Sep 28 2022

			In base 2, 6 is 110 and its first divisors are 1 and 2, that is, 1 and 10.

Cf. A164894, A175252 (base 10), A357429 (base 3).

isok(k) = my(s=[]); fordiv(k, d, s=concat(s, binary(d)); if (fromdigits(s, 2)==k, return(1)); if (fromdigits(s,2)> k, return(0)));

from sympy import divisors
def ok(n):
    target, s = bin(n)[2:], ""
    if target[0] != "1": return False
    for d in divisors(n):
        s += bin(d)[2:]
        if len(s) >= len(target): return s == target
        elif not target.startswith(s): return False
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Oct 01 2022

a(10)-a(11) from Rémy Sigrist, Sep 28 2022

A357429 Numbers whose digit representation in base 3 is equal to the digit representation in base 3 of the initial terms of their sets of divisors in increasing order.

Original entry on oeis.org

1, 48, 50, 333, 438, 448, 734217, 6561081

Offset: 1

Michel Marcus, Sep 28 2022

			In base 3, 48 is 1210 and its first divisors are 1, 2 and 3, that is, 1, 2 and 10.

Cf. A175252 (base 10), A357428 (base 2).

isok(k) = my(s=[]); fordiv(k, d, s=concat(s, digits(d, 3)); if (fromdigits(s, 3)==k, return(1)); if (fromdigits(s, 3)> k, return(0)));

from sympy import divisors
from sympy.ntheory import digits
def ok(n):
    target, s = "".join(map(str, digits(n, 3)[1:])), ""
    if target[0] != "1": return False
    for d in divisors(n):
        s += "".join(map(str, digits(d, 3)[1:]))
        if len(s) >= len(target): return s == target
        elif not target.startswith(s): return False
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Oct 01 2022

A357430 a(n) is the least integer > 1 such that its digit representation in base n is equal to the digit representation in base n of the initial terms of its set of divisors in increasing order.

Original entry on oeis.org

6, 48, 6, 182, 8, 66, 10, 102, 12, 1586, 14, 198, 16, 258, 18, 345, 20, 402, 22, 486, 24, 306484, 26, 678, 28, 786, 30, 26102, 32, 1026, 34, 1158, 36, 1335, 38, 1446, 40, 1602, 42, 204741669824, 44, 1938, 46, 2118, 48, 2355, 50, 2502, 52, 2706, 54, 8199524, 56

Offset: 2

Michel Marcus, Sep 28 2022

Rémy Sigrist, Table of n, a(n) for n = 2..148

Cf. A175252 (base 10), A357428 (base 2), A357429 (base 3).

isok(k, b) = my(s=[]); fordiv(k, d, s=concat(s, digits(d, b)); if (fromdigits(s, b)==k, return(1)); if (fromdigits(s, b)> k, return(0)));
a(n) = my(k=2); while(! isok(k, n), k++); k;

from sympy import divisors
from sympy.ntheory import digits
from itertools import count, islice
def ok(n, b):
    target, s = digits(n, b)[1:], []
    if target[0] != 1: return False
    for d in divisors(n):
        s += digits(d, b)[1:]
        if len(s) >= len(target): return s == target
        elif not target[:len(s)] == s: return False
def a(n):
    return next(i for d in count(1) for i in range(n**d, 2*n**d) if ok(i, n))
print([a(n) for n in range(2, 41)]) # Michael S. Branicky, Oct 05 2022

a(2*n) = 2*n + 2 for any n > 1. - Rémy Sigrist, Sep 29 2022

More terms from Rémy Sigrist, Sep 29 2022

A357481 a(n) is the least integer b such that the digit representation of n in base b is equal to the digit representation in base b of the initial terms of the sets of divisors of n in increasing order, or -1 if no such b exists.

Original entry on oeis.org

2, -1, -1, -1, -1, 2, -1, 6, 6, 8, -1, 10, -1, 12, 12, 14, -1, 16, -1, 18, 18, 20, -1, 22, 20, 24, 24, 26, -1, 28, -1, 30, 30, 32, 30, 34, -1, 36, 36, 38, -1, 40, -1, 42, 42, 44, -1, 3, 42, 3, 48, 2, -1, 52, 50, 54, 54, 56, -1, 58, -1, 60, 2, 62, 60, 7, -1, 66, 66, 68, -1, 70, -1, 72, 7

Offset: 1

Michel Marcus, Sep 30 2022

It appears that a(n) != -1 iff n in A056653.

			In base 10, 12 is a term of A175252, and there is no other lesser base for which 12 works, so a(12) = 10.

Michel Marcus, Table of n, a(n) for n = 1..10000
Peter Munn, Scatterplot with asinh-scale axes. (Axis labeled "A000027(n)" is n.)

Cf. A056653, A175252, A357428, A357429.

isok(k, b) = my(s=[]); fordiv(k, d, s=concat(s, digits(d, b)); if (fromdigits(s, b)==k, return(1)); if (fromdigits(s, b)> k, return(0)));
a(n) = if (n==1, 2, for (b=2, n-1, if (isok(n, b), return(b))); return(-1););

from sympy import divisors
from sympy.ntheory import digits
def ok(n, b):
    target, s = digits(n, b)[1:], []
    if target[0] != 1: return False
    for d in divisors(n):
        s += digits(d, b)[1:]
        if len(s) >= len(target): return s == target
def a(n):
    if n == 1: return 2
    for b in range(2, n):
        if ok(n, b): return b
    return -1
print([a(n) for n in range(1, 76)]) # Michael S. Branicky, Oct 05 2022

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

Original entry on oeis.org

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

Michel Marcus, Sep 22 2022

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

			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.

Cf. A037278, A175252 (a subsequence).
Cf. A004022 (a subsequence).
Subsequence of A131835.

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

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] == ""));

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

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

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

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A357692 Integers k such that A037278(k) is a term of A175252.

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A357428 Numbers whose digit representation in base 2 is equal to the digit representation in base 2 of the initial terms of their sets of divisors in increasing order.

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A357429 Numbers whose digit representation in base 3 is equal to the digit representation in base 3 of the initial terms of their sets of divisors in increasing order.

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A357430 a(n) is the least integer > 1 such that its digit representation in base n is equal to the digit representation in base n of the initial terms of its set of divisors in increasing order.

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A357481 a(n) is the least integer b such that the digit representation of n in base b is equal to the digit representation in base b of the initial terms of the sets of divisors of n in increasing order, or -1 if no such b exists.

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

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