A247047 Numbers k such that k^2 contains exactly 2 distinct digits and k^3 contains exactly 3 distinct digits.

5, 6, 8, 9, 15, 30, 173, 300, 3000, 30000, 300000, 3000000, 30000000, 300000000, 3000000000, 30000000000, 300000000000, 3000000000000, 30000000000000, 300000000000000, 3000000000000000

Offset: 1

Author

Derek Orr, Sep 10 2014

Keywords

• nonn
• base
• more

Comments

Intersection of A016069 and A155146.

This sequence is infinite since 3*10^k is always in this sequence for k > 0.

Is 173 the last term not of the form 3*10^k?

3*10^7 < a(14) <= 3*10^8.

The numbers k such that k^2 contains 2 distinct digits, k^3 contains 3 distinct digits, and k^4 contains 4 distinct digits are conjectured to only be 6, 8, and 15. (Intersection of A016069, A155146, and A155150.)

Examples

			k = 15 is a member of this sequence since 15^2 = 225 contains two distinct digits and 15^3 = 3375 contains three distinct digits.
		

Crossrefs

Cf. A016069, A155146.

Programs

Mathematica

Select[Range[3*10^6],Length[DeleteCases[DigitCount[#^2],0]]==2&&Length[ DeleteCases[ DigitCount[#^3],0]]==3&] (* The program generates the first 12 terms of the sequence. *) (* Harvey P. Dale, Jan 21 2023 *)

PARI

for(n=1,3*10^7,d2=digits(n^2);d3=digits(n^3);if(#vecsort(d2,,8)==2&&#vecsort(d3,,8)==3,print1(n,", ")))

Python

A247047_list = [n for n in range(1,10**6) if len(set(str(n**3))) == 3 and len(set(str(n**2))) == 2]
# Chai Wah Wu, Sep 26 2014

Extensions

a(14)-a(15) from Chai Wah Wu, Sep 26 2014

a(16)-a(18) from Kevin P. Thompson, Jul 01 2022

a(19)-a(21) from Michael S. Branicky, Jun 05 2025