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A346537 Squares that are divisible by the product of their nonzero digits.

%I A346537 #39 Aug 16 2021 22:09:23
%S A346537 1,4,9,36,100,144,400,900,1024,1296,2304,2500,2916,3600,10000,11664,
%T A346537 12100,14400,22500,32400,40000,41616,78400,82944,90000,102400,110224,
%U A346537 121104,122500,129600,152100,176400,186624,200704,202500,219024,230400,250000,260100,291600
%N A346537 Squares that are divisible by the product of their nonzero digits.
%H A346537 Jean-Marie De Koninck and Florian Luca, <a href="https://doi.org/10.4171/PM/1777">Positive integers divisible by the product of their nonzero digits</a>, Port. Math. 64 (2007) 75-85. (This proof for upper bounds contains an error. See the paper below.)
%H A346537 Jean-Marie De Koninck and Florian Luca, <a href="https://doi.org/10.4171/PM/1999">Corrigendum to "Positive integers divisible by the product of their nonzero digits", Portugaliae Math. 64 (2007), 1: 75-85</a>, Port. Math. 74 (2017), 169-170.
%e A346537 For the perfect square 1024 = 32^2 the product of its nonzero digits is 8 which divides 1024.
%t A346537 Select[Range[500]^2, Divisible[#, Times @@ Select[IntegerDigits[#], #1 > 0 &]] &] (* _Amiram Eldar_, Jul 23 2021 *)
%o A346537 (Python)
%o A346537 from math import prod
%o A346537 def nzpd(n): return prod([int(d) for d in str(n) if d != '0'])
%o A346537 def ok(sqr): return sqr > 0 and sqr%nzpd(sqr) == 0
%o A346537 print(list(filter(ok, (i*i for i in range(541))))) # _Michael S. Branicky_, Jul 23 2021
%o A346537 (PARI) isok(m) = issquare(m) && !(m % vecprod(select(x->(x>0), digits(m))));
%o A346537 lista(nn) = for (m=1, nn, if (isok(m^2), print1(m^2, ", "))); \\ _Michel Marcus_, Jul 23 2021
%Y A346537 Intersection of A000290 and A055471.
%K A346537 nonn,base
%O A346537 1,2
%A A346537 _Michael Gohn_, _Joshua Harrington_, _Sophia Lebiere_, _Hani Samamah_, _Kyla Shappell_, _Wing Hong Tony Wong_, Jul 23 2021