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

1,2

Links

Table of n, a(n) for n=1..40.

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.

Example

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

Mathematica

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

Prog

(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
(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

Crossrefs

Intersection of A000290 and A055471.

Sequence in context: A126161 A179934 A239213 * A339999 A018224 A149137

Adjacent sequences: A346534 A346535 A346536 * A346538 A346539 A346540

Keyword

nonn,base

Author

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

Status

approved