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A345397
a(n) is the least number k such that {k, k^2, ..., k^n} are all digitally balanced numbers in base 2 (A031443), or 0 if no such k exists.
4
2, 212, 3274, 15113, 236417, 15975465, 991017155, 4006725713, 4079348720699
OFFSET
1,1
COMMENTS
Conjecture: all a(n) > 0. - Alex Ratushnyak, Apr 26 2022
EXAMPLE
a(1) = 2 since 2 is digitally balanced: its binary representation, 10, has the same number of 0's and 1's.
a(2) = 212 since both 212 and 212^2 are digitally balanced: the binary representation of 212, 11010100, has 4 0's and 4 1's, and the binary representation of 212^2, 1010111110010000, has 8 0's and 8 1's.
MATHEMATICA
balQ[n_] := Module[{d = IntegerDigits[n, 2], m}, EvenQ @ (m = Length @ d) && Count[d, 1] == m/2]; f[k_] := Module[{e = 0, r = k}, While[balQ[r], r *= k; e++]; e]; mx = 5; s = Table[0, {mx}]; c = 0; n = 1; While[c < mx, k = f[n]; Do[If[s[[i]] == 0, s[[i]] = n; c++], {i, 1, k}]; n++]; s
PROG
(Python)
from itertools import count, islice
from sympy.utilities.iterables import multiset_permutations
def isbalanced(n): b = bin(n)[2:]; return b.count("0") == b.count("1")
def A031443gen(): yield from (int("1"+"".join(p), 2) for n in count(1) for p in multiset_permutations("0"*n+"1"*(n-1)))
def a(n):
for k in A031443gen():
if all(isbalanced(k**i) for i in range(2, n+1)):
return k
print([a(n) for n in range(1, 6)]) # Michael S. Branicky, Apr 26 2022
CROSSREFS
KEYWORD
nonn,base,more
AUTHOR
Amiram Eldar, Jun 17 2021
EXTENSIONS
a(9) from Bert Dobbelaere, Jun 18 2021
Name edited by Alex Ratushnyak, Apr 26 2022
STATUS
approved