A357428 -id:A357428 - cp's OEIS frontend

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.

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

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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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