A014569 Super-3 Numbers (3n^3 contains substring '333' in its decimal expansion).

261, 462, 471, 481, 558, 753, 1036, 1046, 1471, 1645, 1752, 1848, 1923, 1926, 1968, 2031, 2231, 2232, 2363, 2395, 2471, 2591, 2610, 3058, 3087, 3148, 3163, 3172, 3181, 3471, 3494, 3542, 3851, 3884, 4143, 4269, 4314, 4471, 4527, 4554, 4620, 4710, 4732

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• base

Comments

For any term a(n), all numbers a(n)*10^k, k >= 0, are also in the sequence. More interestingly, all numbers N == 471 (mod 1000) are in the sequence, since 471^3*3 == 333 (mod 1000). - M. F. Hasler, Jul 16 2024

Conjecture: a(n) ~ n. - Charles R Greathouse IV, Dec 04 2024

Examples

			1752 is in the sequence since 3 * 1752^3 = 161'333'13024.
		

References

• C. A. Pickover, Keys to Infinity. New York: Wiley, p. 7, 1995.

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Giovanni Resta, super-d numbers, personal web site "Numbers Aplenty", 2013.
• Eric Weisstein's World of Mathematics, Super-d Number.

Crossrefs

Cf. A032743-A032749 (similar for d=2, ..., 9).

Programs

Mathematica

Select[Range[5000],MemberQ[Partition[IntegerDigits[3#^3],3,1],{3,3,3}]&] (* Harvey P. Dale, Feb 01 2013 *)

PARI

select( {is_A014569(n, d=3, m=10^d, r=m\9*d)=n=d*n^d; until(r>n\=10, n%m==r && return(1))}, [0..4999]) \\ Using the (optional) 2nd arg d=2..9 allows to compute the sequences A032743-A032749. - M. F. Hasler, Jul 16 2024

Python

is_A014569=lambda n, d=3: str(d)*d in str(d*n**d) # M. F. Hasler, Jul 16 2024

Formula

n < a(n) < 200n for n > 2. - Charles R Greathouse IV, Dec 04 2024

Extensions

Corrected and extended by Patrick De Geest, May 15 1998

Offset changed to 1 by M. F. Hasler, Jul 16 2024