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Squares in A030299.
3

%I #39 Apr 13 2022 07:44:02

%S 1,13527684,34857216,65318724,73256481,81432576,139854276,152843769,

%T 157326849,215384976,245893761,254817369,326597184,361874529,

%U 375468129,382945761,385297641,412739856,523814769,529874361,537219684,549386721,587432169,589324176,597362481,615387249

%N Squares in A030299.

%C Conjecture: there are infinitely many terms.

%D John D. Dixon and Brian Mortimer, Permutation groups. Graduate Texts in Mathematics, 163. Springer-Verlag, New York, 1996. xii+346 pp. ISBN: 0-387-94599-7 MR1409812 (98m:20003).

%H Chai Wah Wu, <a href="/A352329/b352329.txt">Table of n, a(n) for n = 1..3185</a>

%o (Python)

%o from itertools import permutations

%o def pmap(s, m): return sum(s[i-1]*10**(m-i) for i in range(1, len(s)+1))

%o def agen():

%o m = 1

%o while True:

%o for s in permutations(range(1, m+1)): yield pmap(s, m)

%o m += 1

%o def aupton(terms):

%o alst, g = [], agen()

%o while len(alst) < terms: alst += [next(g)]

%o return alst

%o def is_perfect_square(n):

%o return round(n ** 0.5) ** 2 == n

%o print([x for x in aupton(5000000) if is_perfect_square(x)])

%o (Python)

%o from itertools import count, islice, permutations

%o from sympy import integer_nthroot

%o def A352329_gen(): # generator of terms

%o for l in count(1):

%o if (r := l*(l+1)//2 % 9) == 0 or r == 1 or r == 4 or r == 7:

%o m = tuple(10**(l-i-1) for i in range(l))

%o for p in permutations(range(1,l+1)):

%o if integer_nthroot(n := sum(prod(k) for k in zip(m,p)),2)[1]:

%o yield n

%o A352329_list = list(islice(A352329_gen(),10)) # _Chai Wah Wu_, Mar 21-22 2022

%Y Cf. A030299, A240763.

%K nonn,base

%O 1,2

%A _Luca Onnis_ and _Marco RipĂ _, Mar 12 2022