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Super-4 Numbers (4 * n^4 contains substring '4444' in its decimal expansion).
3

%I #29 Jul 17 2024 09:00:12

%S 1168,4972,7423,7752,8431,10267,11317,11487,11549,11680,16588,16664,

%T 16837,18257,18597,19784,19933,22217,22504,22819,22829,24078,24331,

%U 24514,25296,25698,26685,26738,27812,27973,28988,32466,32467,32735,34078,34636,35248,36219,36602

%N Super-4 Numbers (4 * n^4 contains substring '4444' in its decimal expansion).

%C a(17) = 19933 and a(20) = 22819 are such that a(n)^4 == 111121 (mod 10^6), therefore any number ending in (0|5)19933 or in (0|5)22819, where (a|b) means a or b, is in the sequence. Of course, for each term a(n), all numbers a(n)*10^k, k >= 0, are also in the sequence. - _M. F. Hasler_, Jul 16 2024

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

%H Vincenzo Librandi, <a href="/A032744/b032744.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[33000],MemberQ[Partition[IntegerDigits[4*#^4],4,1],{4,4,4,4}]&] (* _Harvey P. Dale_, Mar 22 2012 *)

%o (PARI) select( is_A032744(n)=is_A032743(n,4), [1..33333]) \\ _M. F. Hasler_, Jul 16 2024

%o (Python) is_A032744=lambda n: '4444' in str(4*n**4) # _M. F. Hasler_, Jul 16 2024

%Y Cf. A014569 (d=3), A032743-A032749 (d=2, ..., 9).

%K nonn,base

%O 1,1

%A _Patrick De Geest_, May 15 1998

%E Offset changed to 1 by _M. F. Hasler_, Jul 16 2024