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A045524 Numbers k such that k! has initial digit '5'.

%I A045524 #27 Jan 05 2025 19:51:35
%S A045524 7,21,38,46,61,66,81,119,137,144,150,165,189,196,206,209,221,224,235,
%T A045524 243,248,253,258,279,292,340,342,353,362,383,413,429,440,488,508,529,
%U A045524 540,584,597,611,630,651,662,679,685,704,711,718,725,732,764,782,812
%N A045524 Numbers k such that k! has initial digit '5'.
%C A045524 n such that A000030(A000142(n)) = 5. - _Robert Israel_, Feb 07 2017
%C A045524 The asymptotic density of this sequence is log_10(6/5) = 0.079181... (Kunoff, 1987). - _Amiram Eldar_, Jul 17 2020
%H A045524 Chai Wah Wu, <a href="/A045524/b045524.txt">Table of n, a(n) for n = 1..10000</a>
%H A045524 Sharon Kunoff, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/25-4/kunoff.pdf">N! has the first digit property</a>, The Fibonacci Quarterly, Vol. 25, No. 4 (1987), pp. 365-367.
%H A045524 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A045524 A008905(a(n)) = 5. - _Amiram Eldar_, Jul 17 2020
%e A045524 7 is a term since 7! = 5040 has the initial digit 5.
%p A045524 filter:= proc(t) local tf;
%p A045524 tf:= t!;
%p A045524 floor(tf/10^ilog10(tf)) = 5
%p A045524 end proc:
%p A045524 select(filter, [$1..1000]); # _Robert Israel_, Feb 07 2017
%t A045524 Select[ Range[ 850 ], IntegerDigits[ #! ] [[1]] == 5 & ]
%o A045524 (PARI) isok(n) = digits(n!)[1] == 5; \\ _Michel Marcus_, Feb 08 2017
%Y A045524 For factorials with initial digit d (1 <= d <= 9) see A045509, A045510, A045511, A045516, A045517, A045518, A282021, A045519; A045520, A045521, A045522, A045523, A045525, A045526, A045527, A045528, A045529.
%Y A045524 Cf. A000030, A000142, A008905.
%K A045524 nonn,base
%O A045524 1,1
%A A045524 _Jeff Burch_