A096257 The least k whose n-th root contains k as a string of digits to the immediate right of the decimal point (excluding leading zeros).

8, 2, 3, 633, 19703, 89, 69, 56, 46, 39, 33, 29, 25, 22, 20, 18, 16, 14, 13, 12, 11, 10, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 138, 133, 128, 124, 120, 116, 113, 109, 106, 103, 100, 97, 95, 92, 90, 87, 85, 83, 81, 79, 77, 75, 74, 72, 70, 69, 67, 66, 65, 63, 62, 61, 59, 58, 57

Offset: 2

Author

Paul Lusch and Robert G. Wilson v, Jul 31 2004

Keywords

• base
• nonn

Crossrefs

Cf. A074841, A074762.

Programs

Mathematica

f[k_, n_] := Block[{l = Floor[ Log[10, k] + 1], rd = RealDigits[ k^(1/n), 10, 24], id = IntegerDigits[k]}, rdd = Drop[ rd[[1]], rd[[2]]]; While[ rdd[[1]] == 0, rdd = Drop[rdd, 1]]; Take[rdd, l] == id]; g[n_] := Block[{k = 2}, While[IntegerQ[k^(1/n)] || f[k, n] == False, k++ ]; k]; Table[ g[n], {n, 2, 72}]

Python

import re
from sympy import perfect_power
from decimal import *
getcontext().prec = 24
def lzs(s): return len(s) - 2 - len(s[2:].lstrip('0')) # # of leading zeros
def cond(sk, sroot, k, n): # is condition true, with precision verification
    if perfect_power(k, [n]): return False # decimal part should be all 0's
    assert lzs(sroot) + len(sk) < len(sroot) - 3, (n, "increase precision")
    return re.match("0.0*"+sk, sroot)
def a(n):
    k, power = 1, Decimal(1)/Decimal(n)
    rootk, sk = Decimal(k)**power, str(k)
    while not cond(sk, str(rootk - int(rootk)), k, n):
        k += 1
        rootk, sk = Decimal(k)**power, str(k)
    return k
print([a(n) for n in range(2, 73)]) # Michael S. Branicky, Aug 02 2021