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

%I A032743 #39 Feb 16 2025 08:32:36
%S A032743 19,31,69,81,105,106,107,119,127,131,169,181,190,219,231,247,269,281,
%T A032743 310,318,319,331,332,333,334,335,336,337,338,339,348,369,381,419,431,
%U A032743 454,469,481,511,519,531,558,569,581,601,619,631,669,679,681,690,715
%N A032743 Super-2 Numbers (2 * n^2 contains substring '22' in its decimal expansion).
%C A032743 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 A032743 Conjecture: a(n) ~ n. - _Charles R Greathouse IV_, Dec 04 2024
%D A032743 C. A. Pickover, "Keys to Infinity", New York: Wiley, p. 7, 1995.
%H A032743 Harvey P. Dale, <a href="/A032743/b032743.txt">Table of n, a(n) for n = 1..1000</a>
%H A032743 Giovanni Resta, <a href="https://www.numbersaplenty.com/set/super-d_number/">super-d numbers</a>, personal web site "Numbers Aplenty", 2013
%H A032743 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Super-dNumber.html">Super-d Number.</a>
%t A032743 Select[Range[1000],MemberQ[Partition[IntegerDigits[2#^2],2,1],{2,2}]&] (* _Harvey P. Dale_, May 09 2012 *)
%t A032743 Select[Range[750],SequenceCount[IntegerDigits[2#^2],{2,2}]>0&] (* _Harvey P. Dale_, May 13 2022 *)
%o A032743 (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 A032743 (Python) is_A032743=lambda n, d=2: str(d)*d in str(d*n**d) # _M. F. Hasler_, Jul 16 2024
%Y A032743 Cf. A014569 (similar for d=3), A032744 - A032749 (similar for d=4, ..., 9).
%K A032743 nonn,base
%O A032743 1,1
%A A032743 _Patrick De Geest_, May 15 1998