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A339999 Squares that are divisible by both the sum of their digits and the product of their nonzero digits.

%I A339999 #120 Jan 05 2025 19:51:41
%S A339999 1,4,9,36,100,144,400,900,1296,2304,2916,3600,10000,11664,12100,14400,
%T A339999 22500,32400,40000,41616,82944,90000,121104,122500,129600,152100,
%U A339999 176400,186624,202500,219024,230400,260100,291600,360000,419904,435600,504100
%N A339999 Squares that are divisible by both the sum of their digits and the product of their nonzero digits.
%H A339999 H. G. Grundman, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/32-2/grundman.pdf">Sequences of consecutive n-Niven numbers</a>, Fibonacci Quarterly (1994) 32 (2): 174-175.
%H A339999 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 A339999 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 A339999 For the perfect square 144 = 12^2, the sum of its digits is 9, which divides 144, and the product of its nonzero digits is 16, which also divides 144 so 144 is a term of the sequence.
%t A339999 Select[Range[720]^2, And @@ Divisible[#, {Plus @@ (d = IntegerDigits[#]), Times @@ Select[d, #1 > 0 &]}] &] (* _Amiram Eldar_, Jul 23 2021 *)
%o A339999 (Python)
%o A339999 from math import prod
%o A339999 def sumd(n): return sum(map(int, str(n)))
%o A339999 def nzpd(n): return prod([int(d) for d in str(n) if d != '0'])
%o A339999 def ok(sqr): return sqr > 0 and sqr%sumd(sqr) == 0 and sqr%nzpd(sqr) == 0
%o A339999 print(list(filter(ok, (i*i for i in range(1001)))))
%o A339999 # _Michael S. Branicky_, Jul 23 2021
%Y A339999 Intersection of A000290, A005349 and A055471.
%Y A339999 Cf. A118547, A342262, A342650.
%K A339999 nonn,base
%O A339999 1,2
%A A339999 _Michael Gohn_, _Joshua Harrington_, _Sophia Lebiere_, _Hani Samamah_, _Kyla Shappell_, _Wing Hong Tony Wong_, Jul 23 2021