A032743 - OEIS

%I #38 Dec 04 2024 09:50:21

%S 19,31,69,81,105,106,107,119,127,131,169,181,190,219,231,247,269,281,

%T 310,318,319,331,332,333,334,335,336,337,338,339,348,369,381,419,431,

%U 454,469,481,511,519,531,558,569,581,601,619,631,669,679,681,690,715

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

%C 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

%C Conjecture: a(n) ~ n. - _Charles R Greathouse IV_, Dec 04 2024

%D C. A. Pickover, "Keys to Infinity", New York: Wiley, p. 7, 1995.

%H Harvey P. Dale, <a href="/A032743/b032743.txt">Table of n, a(n) for n = 1..1000</a>

%H Giovanni Resta, <a href="https://www.numbersaplenty.com/set/super-d_number/">super-d numbers</a>, personal web site "Numbers Aplenty", 2013

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Super-dNumber.html">Super-d Number.</a>

%t Select[Range[1000],MemberQ[Partition[IntegerDigits[2#^2],2,1],{2,2}]&] (* _Harvey P. Dale_, May 09 2012 *)

%t Select[Range[750],SequenceCount[IntegerDigits[2#^2],{2,2}]>0&] (* _Harvey P. Dale_, May 13 2022 *)

%o (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

%o (Python) is_A032743=lambda n, d=2: str(d)*d in str(d*n**d) # _M. F. Hasler_, Jul 16 2024

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

%K nonn,base

%O 1,1

%A _Patrick De Geest_, May 15 1998