A030290 a(n) is the smallest k > a(n-1) such that k^3 has no digit in common with a(n-1)^3.

0, 1, 2, 3, 4, 5, 7, 8, 16, 18, 40, 45, 67, 98, 150, 204, 237, 44216, 46443, 78742, 79930, 130714, 173000, 185604, 1000000, 1304963, 10000000, 13049563, 100000000, 130495593, 1000000000, 1304955895, 10000000000, 13049558812, 100000000000, 130495588186, 1000000000000, 1304955880707

Offset: 0

Author

Patrick De Geest

Keywords

• nonn
• base

Comments

From a(24) = 10^6 on, we have a(2k) = 10^(k-6) and a(2k+1) ~ c*a(2k) with c = (20/9)^(1/3) = 1.30495588... Indeed, a(2k)^3 = 1000^(k-6) has then only digits 0 and 1, and the next term must have a cube >= 2.2222...*1000^(k-6), so a(2k+1) will be the cube root of the next larger cube with no digit 0 and 1. - M. F. Hasler, Nov 12 2017

Examples

			a(5) = 5 and 5^3 = 125 has no digit in common with the cube of a(4) = 4, 4^3 = 64.
But a(6) cannot be equal to 6, because 6^3 = 216 has digits '1' and '2' in common with 5^3 = 125.
		

Links

• David W. Wilson, Table of n, a(n) for n = 0..90

Crossrefs

Cf. A030289.

Programs

PARI

next_A030290(n, S=Set(digits(n^3)))={if(n>18e4,S[1]&&return(10^logint(n<<3,10));n\=sqrtn(.45,3));for(k=n+1,oo, #setintersect(Set(digits(k^3)), S)||return(k))} \\ M. F. Hasler, Nov 12 2017
print1(a=0); for(i=1, 99, print1(", "a=next_A030290(a))) \\ M. F. Hasler, Nov 08 2017

Formula

a(n) = A030289(n)^(1/3). - David W. Wilson, Nov 08 2017

For k >= 12, a(2k) = 10^(k-6), and a(2k+1) > c*a(2k) with approximate equality, where c = (20/9)^(1/3) = 1.30495588... - M. F. Hasler, Nov 12 2017