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a(n)^2 is the least square that has exactly n 0's in base n.
3

%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