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

1, 4, 9, 36, 100, 144, 400, 900, 1024, 1296, 2304, 2500, 2916, 3600, 10000, 11664, 12100, 14400, 22500, 32400, 40000, 41616, 78400, 82944, 90000, 102400, 110224, 121104, 122500, 129600, 152100, 176400, 186624, 200704, 202500, 219024, 230400, 250000, 260100, 291600

Offset: 1

Author

Michael Gohn, Joshua Harrington, Sophia Lebiere, Hani Samamah, Kyla Shappell, Wing Hong Tony Wong, Jul 23 2021

Keywords

• nonn
• base

Examples

			For the perfect square 1024 = 32^2 the product of its nonzero digits is 8 which divides 1024.
		

Links

• Jean-Marie De Koninck and Florian Luca, Positive integers divisible by the product of their nonzero digits, Port. Math. 64 (2007) 75-85. (This proof for upper bounds contains an error. See the paper below.)
• Jean-Marie De Koninck and Florian Luca, Corrigendum to "Positive integers divisible by the product of their nonzero digits", Portugaliae Math. 64 (2007), 1: 75-85, Port. Math. 74 (2017), 169-170.

Crossrefs

Intersection of A000290 and A055471.

Programs

Mathematica

Select[Range[500]^2, Divisible[#, Times @@ Select[IntegerDigits[#], #1 > 0 &]] &] (* Amiram Eldar, Jul 23 2021 *)

PARI

isok(m) = issquare(m) && !(m % vecprod(select(x->(x>0), digits(m))));
lista(nn) = for (m=1, nn, if (isok(m^2), print1(m^2, ", "))); \\ Michel Marcus, Jul 23 2021

Python

from math import prod
def nzpd(n): return prod([int(d) for d in str(n) if d != '0'])
def ok(sqr): return sqr > 0 and sqr%nzpd(sqr) == 0
print(list(filter(ok, (i*i for i in range(541))))) # Michael S. Branicky, Jul 23 2021