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A352329 Squares in A030299.

%I A352329 #39 Apr 13 2022 07:44:02
%S A352329 1,13527684,34857216,65318724,73256481,81432576,139854276,152843769,
%T A352329 157326849,215384976,245893761,254817369,326597184,361874529,
%U A352329 375468129,382945761,385297641,412739856,523814769,529874361,537219684,549386721,587432169,589324176,597362481,615387249
%N A352329 Squares in A030299.
%C A352329 Conjecture: there are infinitely many terms.
%D A352329 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 A352329 Chai Wah Wu, <a href="/A352329/b352329.txt">Table of n, a(n) for n = 1..3185</a>
%o A352329 (Python)
%o A352329 from itertools import permutations
%o A352329 def pmap(s, m): return sum(s[i-1]*10**(m-i) for i in range(1, len(s)+1))
%o A352329 def agen():
%o A352329   m = 1
%o A352329   while True:
%o A352329     for s in permutations(range(1, m+1)): yield pmap(s, m)
%o A352329     m += 1
%o A352329 def aupton(terms):
%o A352329   alst, g = [], agen()
%o A352329   while len(alst) < terms: alst += [next(g)]
%o A352329   return alst
%o A352329 def is_perfect_square(n):
%o A352329         return round(n ** 0.5) ** 2 == n
%o A352329 print([x for x in aupton(5000000) if is_perfect_square(x)])
%o A352329 (Python)
%o A352329 from itertools import count, islice, permutations
%o A352329 from sympy import integer_nthroot
%o A352329 def A352329_gen(): # generator of terms
%o A352329     for l in count(1):
%o A352329         if (r := l*(l+1)//2 % 9) == 0 or r == 1 or r == 4 or r == 7:
%o A352329             m = tuple(10**(l-i-1) for i in range(l))
%o A352329             for p in permutations(range(1,l+1)):
%o A352329                 if integer_nthroot(n := sum(prod(k) for k in zip(m,p)),2)[1]:
%o A352329                     yield n
%o A352329 A352329_list = list(islice(A352329_gen(),10)) # _Chai Wah Wu_, Mar 21-22 2022
%Y A352329 Cf. A030299, A240763.
%K A352329 nonn,base
%O A352329 1,2
%A A352329 _Luca Onnis_ and _Marco Ripà_, Mar 12 2022