A032743 Super-2 Numbers (2 * n^2 contains substring '22' in its decimal expansion).

19, 31, 69, 81, 105, 106, 107, 119, 127, 131, 169, 181, 190, 219, 231, 247, 269, 281, 310, 318, 319, 331, 332, 333, 334, 335, 336, 337, 338, 339, 348, 369, 381, 419, 431, 454, 469, 481, 511, 519, 531, 558, 569, 581, 601, 619, 631, 669, 679, 681, 690, 715

Offset: 1

Author

Patrick De Geest, May 15 1998

Keywords

• nonn
• base

Comments

For any term a(n), all numbers a(n)*10^k, k >= 0, are also in the sequence. Moreover, the first four terms satisfy 2*a(n)^2 == 22 (mod 100), therefore any number ending in 19, 31, 69 or 81 (possibly followed by trailing '0's) is in the sequence. - M. F. Hasler, Jul 16 2024

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

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. A014569 (similar for d=3), A032744 - A032749 (similar for d=4, ..., 9).

Programs

Mathematica

Select[Range[1000],MemberQ[Partition[IntegerDigits[2#^2],2,1],{2,2}]&] (* Harvey P. Dale, May 09 2012 *)
Select[Range[750],SequenceCount[IntegerDigits[2#^2],{2,2}]>0&] (* Harvey P. Dale, May 13 2022 *)

PARI

select( {is_A032743(n, d=2, m=10^d, r=m\9*d)=n=d*n^d; until(r>n\=10, n%m==r && return(1))}, [0..999]) \\ M. F. Hasler, Jul 16 2024

Python

is_A032743=lambda n, d=2: str(d)*d in str(d*n**d) # M. F. Hasler, Jul 16 2024