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Numbers k such that the decimal expansion of k! begins with k.
2

%I #27 Aug 07 2021 01:27:40

%S 1,2,198,217,5936,7400,8999,11672,5814279,9242361,10430012,21603226,

%T 28563733,252544448,5042264464,5795685640,421559495895

%N Numbers k such that the decimal expansion of k! begins with k.

%C The inclusion of the deleted term k=171489648 seems likely to have resulted from a precision problem: computing as an intermediate result x = log_10(k!) = 1337609557.23423787495... and storing that result rounded to its nearest IEEE 754 double-precision binary floating-point number D = 5610341116345793 / 2^22 would result in floor(10^(D-1337609549)) = 171489648 = k, whereas floor(10^(x-1337609549)) = 171489634. - _Jon E. Schoenfield_, Nov 07 2014

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>

%e From _Jon E. Schoenfield_, Nov 07 2014: (Start)

%e 198! = 198.155...*10^368

%e 217! = 217.443...*10^412

%e 5936! = 5936.670...*10^19820

%e 7400! = 7400.662...*10^25417

%e 8999! = 8999.544...*10^31674

%e 11672! = 11672.612...*10^42401

%e 5814279! = 5814279.634...*10^36805554

%e 9242361! = 9242361.412...*10^60366371

%e 10430012! = 10430012.795...*10^68671095

%e 21603226! = 21603226.750...*10^149067098

%e 28563733! = 28563733.968...*10^200560833

%e 252544448! = 252544448.239...*10^2012285104

%e 5042264464! = 5042264464.601...*10^46733376736

%e 5795685640! = 5795685640.984...*10^54066852117

%e 421559495895! = 421559495895.661...*10^4717488684964

%e (End)

%t Do[ If[ RealDigits[ n ][ [ 1 ] ] == Take[ RealDigits[ n! ][ [ 1 ] ], RealDigits[ n ][ [ 2 ] ] ], Print[ n ] ], {n, 1, 10^5} ]

%Y Cf. A000142.

%K hard,nonn,base

%O 1,2

%A _Jeff Burch_

%E More terms from _David W. Wilson_

%E Incorrect term 171489648 deleted and terms a(13)-a(17) added by _Jon E. Schoenfield_, Nov 07 2014