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%I #29 Apr 09 2021 09:22:35
%S 2,24,16,280,216,3430,4096,19683,100000,4348377,2985984,154457888,
%T 105413504,4442343750,4294967296,313909084845,198359290368,
%U 8712567840033,10240000000000,500396429346030,584318301411328,38112390316557080,36520347436056576,298023223876953125
%N a(n)^2 is the least square that has exactly n 0's in base n.
%H Chai Wah Wu, <a href="/A342545/b342545.txt">Table of n, a(n) for n = 2..702</a>
%F a(2*n) = A062971(n) = 2*A193678(n).
%e n a(n) a(n)^2 in base n
%e 2 2 4 100
%e 3 24 576 210100
%e 4 16 256 10000
%e 5 280 78400 10002100
%e 6 216 46656 1000000
%e 7 3430 11764900 202000000
%e 8 4096 16777216 100000000
%e 9 19683 387420489 1000000000
%e 10 100000 10000000000 10000000000
%e 11 4348377 18908382534129 6030000000000
%e 12 2985984 8916100448256 1000000000000
%o (PARI) for(b=2,12,for(k=1,oo,my(s=k^2,v=digits(s,b));if(sum(k=1,#v,v[k]==0)==b,print1(k,", ");break)))
%o (Python)
%o from numba import njit
%o @njit # works with 64 bits through a(14)
%o def digits0(n, b):
%o count0 = 0
%o while n >= b:
%o n, r = divmod(n, b)
%o count0 += (r==0)
%o return count0 + (n==0)
%o from sympy import integer_nthroot
%o def a(n):
%o an = integer_nthroot(n**n, 2)[0]
%o while digits0(an*an, n) != n: an += 1
%o return an
%o print([a(n) for n in range(2, 13)]) # _Michael S. Branicky_, Apr 07 2021
%o (Python)
%o from itertools import product
%o from functools import reduce
%o from sympy.utilities.iterables import multiset_permutations
%o from sympy import integer_nthroot
%o def A342545(n):
%o for a in range(1,n):
%o p, q = integer_nthroot(a*n**n,2)
%o if q: return p
%o l = 1
%o while True:
%o cmax = n**(l+n+1)
%o for a in range(1,n):
%o c = cmax
%o for b in product(range(1,n),repeat=l):
%o for d in multiset_permutations((0,)*n+b):
%o p, q = integer_nthroot(reduce(lambda c, y: c*n+y, [a]+d),2)
%o if q: c = min(c,p)
%o if c < cmax:
%o return c
%o l += 1 # _Chai Wah Wu_, Apr 07 2021
%Y Cf. A000290, A062971, A193678, A342260, A342546.
%K nonn,base
%O 2,1
%A _Hugo Pfoertner_, Apr 07 2021
%E More terms from _Chai Wah Wu_, Apr 07 2021
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