A328780 Nonnegative integers k such that k and k^2 have the same number of nonzero digits.

0, 1, 2, 3, 10, 20, 30, 100, 200, 245, 247, 249, 251, 253, 283, 300, 448, 548, 949, 1000, 1249, 1253, 1416, 1747, 1749, 1751, 1753, 1755, 2000, 2245, 2247, 2249, 2251, 2253, 2429, 2450, 2451, 2470, 2490, 2498, 2510, 2530, 2647, 2830, 3000, 3747, 3751, 4480, 4899

Offset: 1

Author

Bernard Schott, Oct 27 2019

Keywords

• nonn
• base

Comments

The idea of this sequence comes from the 1st problem of the 28th British Mathematical Olympiad in 1992 (see the link).

This sequence is infinite because the family of integers {10^k, k >= 0} (A011557) belongs to this sequence.

The numbers m, m + 1, m + 2 where m = 49*10^k - 3, or m = 99*10^k - 3, k >= 3 are terms with all nonzero digits. - Marius A. Burtea, Dec 21 2020

Examples

			247^2 = 61009, hence 247 and 61009 both have 3 nonzero digits, 247 is a term.
		

References

• A. Gardiner, The Mathematical Olympiad Handbook: An Introduction to Problem Solving, Oxford University Press, 1997, reprinted 2011, Pb 1 pp. 57 and 109 (1992)

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000.
• British Mathematical Olympiad, 1992 - Problem 1.
• Index to sequences related to Olympiads.

Crossrefs

Subsequences: A011557, A093136, A093138.

Cf. A052040, A104315, A134844.

Cf. A328781, A328782, A328783.

Programs

Magma

nz:=func; [k:k in [0..5000] | nz(k) eq nz(k^2)]; // Marius A. Burtea, Dec 21 2020

Maple

q:= n->(f->f(n)=f(n^2))(t->nops(subs(0=[][], convert(t, base, 10)))):
select(q, [$0..5000])[];  # Alois P. Heinz, Oct 27 2019

Mathematica

Select[Range[0, 5000], Equal @@ Total /@ Sign@ IntegerDigits[{#, #^2}] &] (* Giovanni Resta, Feb 27 2020 *)

PARI

isok(k) = hammingweight(digits(k)) == hammingweight(digits(k^2)); \\ Michel Marcus, Dec 22 2020

Extensions

More terms from Alois P. Heinz, Oct 27 2019