A045529 -id:A045529 - cp's OEIS frontend

A045521 Numbers k such that k! has initial digit '2'.

2, 4, 16, 20, 23, 30, 32, 34, 39, 44, 47, 54, 58, 68, 75, 85, 86, 87, 113, 114, 115, 126, 135, 142, 148, 153, 159, 163, 168, 172, 180, 184, 188, 195, 205, 208, 211, 214, 217, 220, 223, 226, 234, 237, 242, 247, 252, 257, 262, 269, 276, 278, 287, 289, 302, 304

Offset: 1

Jeff Burch

The asymptotic density of this sequence is log_10(3/2) = 0.176091... (see A154580) (Kunoff, 1987). - Amiram Eldar, Jul 17 2020

			4 is a term since 4! = 24 has the initial digit 2.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Sharon Kunoff, N! has the first digit property, The Fibonacci Quarterly, Vol. 25, No. 4 (1987), pp. 365-367.
Index entries for sequences related to factorial numbers

For factorials with initial digit d (1 <= d <= 9) see A045509, A045510, A045511, A045516, A045517, A045518, A282021, A045519; A045520, A045521, A045522, A045523, A045524, A045525, A045526, A045527, A045528, A045529. See also A000142, A008905, A154580.

Select[ Range[ 310 ], IntegerDigits[ #! ] [ [ 1 ] ] == 2 & ]

isok(n) = digits(n!)[1] == 2; \\ Michel Marcus, Feb 07 2017

A008905(a(n)) = 2. - Amiram Eldar, Jul 17 2020

Offset changed to 1 by Chai Wah Wu, Feb 07 2017

A045522 Numbers k such that k! has initial digit '3'.

Original entry on oeis.org

9, 10, 11, 17, 28, 36, 41, 50, 62, 67, 74, 83, 84, 116, 117, 127, 128, 136, 143, 149, 154, 164, 173, 177, 181, 192, 199, 202, 229, 245, 250, 255, 260, 267, 274, 283, 285, 296, 298, 300, 332, 334, 336, 349, 358, 360, 367, 374, 381, 386, 411, 416, 424, 435, 449

Offset: 1

Jeff Burch

The asymptotic density of this sequence is log_10(4/3) = 0.124938... (Kunoff, 1987). - Amiram Eldar, Jul 17 2020

			9 is a term since 9! = 362880 has the initial digit 3.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Sharon Kunoff, N! has the first digit property, The Fibonacci Quarterly, Vol. 25, No. 4 (1987), pp. 365-367.
Index entries for sequences related to factorial numbers

For factorials with initial digit d (1 <= d <= 9) see A045509, A045510, A045511, A045516, A045517, A045518, A282021, A045519; A045520, A045521, A045522, A045523, A045524, A045525, A045526, A045527, A045528, A045529. See also A000142, A008905.

Select[ Range[ 500 ], IntegerDigits[ #! ] [ [ 1 ] ] == 3 & ]

isok(n) = digits(n!)[1] == 3; \\ Michel Marcus, Feb 07 2017

A008905(a(n)) = 3. - Amiram Eldar, Jul 17 2020

A045523 Numbers k such that k! has initial digit '4'.

Original entry on oeis.org

8, 12, 26, 53, 57, 73, 82, 118, 129, 155, 160, 169, 185, 212, 215, 218, 232, 240, 265, 272, 281, 294, 338, 351, 369, 376, 388, 393, 398, 403, 408, 421, 432, 443, 446, 482, 485, 498, 515, 522, 533, 544, 548, 576, 580, 593, 602, 616, 621, 641, 646, 657, 668

Offset: 1

Jeff Burch

The asymptotic density of this sequence is log_10(5/4) = 0.096910... (Kunoff, 1987). - Amiram Eldar, Jul 17 2020

			8 is a term since 8! = 40320 has the initial digit 4.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Sharon Kunoff, N! has the first digit property, The Fibonacci Quarterly, Vol. 25, No. 4 (1987), pp. 365-367.
Index entries for sequences related to factorial numbers

For factorials with initial digit d (1 <= d <= 9) see A045509, A045510, A045511, A045516, A045517, A045518, A282021, A045519; A045520, A045521, A045522, A045523, A045524, A045525, A045526, A045527, A045528, A045529. See also A000142, A008905.

Select[ Range[ 700 ], IntegerDigits[ #! ] [[ 1 ]] == 4 & ]

isok(n) = digits(n!)[1] == 4; \\ Michel Marcus, Feb 07 2017

A008905(a(n)) = 4. - Amiram Eldar, Jul 17 2020

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

Original entry on oeis.org

7, 21, 38, 46, 61, 66, 81, 119, 137, 144, 150, 165, 189, 196, 206, 209, 221, 224, 235, 243, 248, 253, 258, 279, 292, 340, 342, 353, 362, 383, 413, 429, 440, 488, 508, 529, 540, 584, 597, 611, 630, 651, 662, 679, 685, 704, 711, 718, 725, 732, 764, 782, 812

Offset: 1

Jeff Burch

n such that A000030(A000142(n)) = 5. - Robert Israel, Feb 07 2017

The asymptotic density of this sequence is log_10(6/5) = 0.079181... (Kunoff, 1987). - Amiram Eldar, Jul 17 2020

			7 is a term since 7! = 5040 has the initial digit 5.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Sharon Kunoff, N! has the first digit property, The Fibonacci Quarterly, Vol. 25, No. 4 (1987), pp. 365-367.
Index entries for sequences related to factorial numbers

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.

Cf. A000030, A000142, A008905.

filter:= proc(t) local tf;
tf:= t!;
floor(tf/10^ilog10(tf)) = 5
end proc:
select(filter, [$1..1000]); # Robert Israel, Feb 07 2017

Select[ Range[ 850 ], IntegerDigits[ #! ] [[1]] == 5 & ]

isok(n) = digits(n!)[1] == 5; \\ Michel Marcus, Feb 08 2017

A008905(a(n)) = 5. - Amiram Eldar, Jul 17 2020

A045525 Numbers k such that k! has initial digit '6'.

Original entry on oeis.org

3, 13, 18, 24, 43, 49, 72, 120, 130, 138, 174, 178, 182, 193, 203, 227, 263, 270, 290, 309, 311, 313, 315, 317, 319, 321, 355, 364, 371, 378, 390, 395, 400, 405, 418, 426, 437, 457, 460, 463, 466, 469, 491, 501, 511, 518, 525, 536, 555, 559, 563, 567, 571

Offset: 1

Jeff Burch

The asymptotic density of this sequence is log_10(7/6) = 0.066946... (Kunoff, 1987). - Amiram Eldar, Jul 17 2020

			3 is a term since 3! = 6 has the initial digit 6.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Sharon Kunoff, N! has the first digit property, The Fibonacci Quarterly, Vol. 25, No. 4 (1987), pp. 365-367.
Index entries for sequences related to factorial numbers

For factorials with initial digit d (1 <= d <= 9) see A045509, A045510, A045511, A045516, A045517, A045518, A282021, A045519; A045520, A045521, A045522, A045523, A045524, A045525, A045526, A045527, A045528, A045529. See also A000142, A008905.

Select[ Range[ 600 ], IntegerDigits[ #! ] [ [1] ] == 6 & ]

isok(n) = digits(n!)[1] == 6; \\ Michel Marcus, Feb 08 2017

A008905(a(n)) = 6. - Amiram Eldar, Jul 17 2020

A045526 Numbers k such that k! has initial digit '7'.

Original entry on oeis.org

6, 56, 80, 156, 161, 170, 186, 200, 230, 238, 277, 288, 305, 307, 323, 325, 327, 344, 385, 410, 451, 454, 472, 475, 504, 532, 547, 551, 575, 592, 601, 615, 645, 661, 697, 710, 724, 731, 790, 800, 811, 822, 848

Offset: 1

Jeff Burch

The asymptotic density of this sequence is log_10(8/7) = 0.057991... (Kunoff, 1987). - Amiram Eldar, Jul 17 2020

			6 is a term since 6! = 720 has the initial digit 7.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Sharon Kunoff, N! has the first digit property, The Fibonacci Quarterly, Vol. 25, No. 4 (1987), pp. 365-367.
Index entries for sequences related to factorial numbers

For factorials with initial digit d (1 <= d <= 9) see A045509, A045510, A045511, A045516, A045517, A045518, A282021, A045519; A045520, A045521, A045522, A045523, A045524, A045525, A045526, A045527, A045528, A045529. See also A000142, A008905.

Select[ Range[ 900 ], IntegerDigits[ #! ] [[1]] == 7 & ]

isok(n) = digits(n!)[1] == 7; \\ Michel Marcus, Feb 08 2017

A008905(a(n)) = 7. - Amiram Eldar, Jul 17 2020

A045527 Numbers k such that k! has initial digit '8'.

Original entry on oeis.org

14, 29, 31, 33, 40, 52, 60, 65, 71, 79, 121, 131, 145, 151, 246, 251, 256, 286, 303, 329, 346, 357, 373, 423, 434, 448, 478, 494, 543, 579, 610, 650, 678, 684, 703, 738, 746, 754, 771, 780, 810, 834, 847

Offset: 1

Jeff Burch

The asymptotic density of this sequence is log_10(9/8) = 0.051152... (Kunoff, 1987). - Amiram Eldar, Jul 17 2020

			14 is a term since 14! = 87178291200 has the initial digit 8.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Sharon Kunoff, N! has the first digit property, The Fibonacci Quarterly, Vol. 25, No. 4 (1987), pp. 365-367.
Index entries for sequences related to factorial numbers

For factorials with initial digit d (1 <= d <= 9) see A045509, A045510, A045511, A045516, A045517, A045518, A282021, A045519; A045520, A045521, A045522, A045523, A045524, A045525, A045526, A045527, A045528, A045529. See also A000142, A008905.

Select[ Range[ 900 ], IntegerDigits[ #! ] [ [1] ] == 8 & ]

isok(n) = digits(n!)[1] == 8; \\ Michel Marcus, Feb 08 2017

A008905(a(n)) = 8. - Amiram Eldar, Jul 17 2020

A045528 Numbers k such that k! has initial digit '9'.

Original entry on oeis.org

96, 97, 98, 99, 100, 101, 102, 103, 122, 139, 166, 190, 233, 241, 261, 268, 301, 331, 366, 380, 415, 431, 445, 481, 497, 514, 521, 583, 596, 624, 629, 634, 655, 672, 690, 716, 723, 762, 789, 799, 821, 833, 861, 897

Offset: 1

Jeff Burch

The asymptotic density of this sequence is log_10(10/9) = 1 - log_10(9) = 0.045757... (A104140) (Kunoff, 1987). - Amiram Eldar, Jul 17 2020

			96 is a term since 96! = 9.916779... * 10^149 has the initial digit 9.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Sharon Kunoff, N! has the first digit property, The Fibonacci Quarterly, Vol. 25, No. 4 (1987), pp. 365-367.
Index entries for sequences related to factorial numbers

For factorials with initial digit d (1 <= d <= 9) see A045509, A045510, A045511, A045516, A045517, A045518, A282021, A045519; A045520, A045521, A045522, A045523, A045524, A045525, A045526, A045527, A045528, A045529. See also A000142, A008905, A104140.

Select[ Range[ 900 ], IntegerDigits[ #! ] [ [1] ] == 9 & ]

isok(n) = digits(n!)[1] == 9; \\ Michel Marcus, Feb 08 2017

A008905(a(n)) = 9. - Amiram Eldar, Jul 17 2020

More terms from Robert G. Wilson v, Jan 03 2001

A282021 Factorials with initial digit '7'.

Original entry on oeis.org

720, 710998587804863451854045647463724949736497978881168458687447040000000000000, 71569457046263802294811533723186532165584657342365752577109445058227039255480148842668944867280814080000000000000000000

Offset: 1

N. J. A. Sloane, Feb 07 2017

Benford's law shows that this sequence will contain about (log(8) - log(7))/log(10) =~ 5.8% of all factorials.

Vincenzo Librandi, Table of n, a(n) for n = 1..22
Index entries for sequences related to Benford's law

For factorials with initial digit d (1 <= d <= 9) see A045509, A045510, A045511, A045516, A045517, A045518, A282021, A045519; A045520, A045521, A045522, A045523, A045524, A045525, A045526, A045527, A045528, A045529. See also A000142.

Select[Range[100]!, First[IntegerDigits[#]] == 7 &] (* Vincenzo Librandi, Feb 08 2017 *)

a(n) = A000142(A045526(n)). - Amiram Eldar, Jul 19 2020

A001999 a(n) = a(n-1)*(a(n-1)^2 - 3).

Original entry on oeis.org

3, 18, 5778, 192900153618, 7177905237579946589743592924684178

Offset: 0

N. J. A. Sloane

The next terms in the sequence contain 102 and 305 digits. - Harvey P. Dale, Jun 09 2011
From Peter Bala, Nov 13 2012: (Start)
The present sequence is the case x = 3 of the following general remarks. For other cases see A219160 (x = 4), A219161 (x = 5) and A112845 (x = 6).
Let x > 2 and let alpha := {x + sqrt(x^2 - 4)}/2. Define a sequence a(n) (which depends on x) by setting a(n) = alpha^(3^n) + (1/alpha)^(3^n). Then it is easy to verify that the sequence a(n) satisfies the recurrence equation a(n+1) = a(n)^3 - 3*a(n) with the initial condition a(0) = x.
We have the following identity, valid for x > 2: sqrt((x + 2)/(x - 2)) = (1 + 2/(x-1))*sqrt((y + 2)/(y - 2)), where y = x^3 - 3*x. Iterating the identity produces the product expansion sqrt((x+2)/(x-2)) = Product_{n = 0..oo} (1 + 2/(a(n) - 1)), with a(0) = x and a(n+1) = a(n)^3 - 3*a(n). The rate of convergence is cubic (Fine).
For similar results to the above see A001566 and A219162. (End)
Let b(n) = a(n) - 3. The sequence {b(n)} appears to be a strong divisibility sequence, that is, gcd(b(n),b(m)) = b(gcd(n,m)) for n, m >= 1. - Peter Bala, Dec 08 2022

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..7
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
E. B. Escott, Rapid method for extracting a square root, Amer. Math. Monthly, Vol. 44, No. 10 (1937), pp. 644-646.
N. J. Fine, Infinite products for k-th roots, Amer. Math. Monthly Vol. 84, No. 8 (Oct. 1977), pp. 629-630.
Walther Janous, Problem B-916, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 39, No. 2 (2001), p. 181; Subscript Is Power, Solution to Problem B-916 by H.-J. Seiffert, ibid., Vol. 40, No. 1 (2002), p. 86.
Hideyuki Ohtsu, Problem B-1316, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 60, No. 4 (2022), p. 365.
Eric Weisstein's World of Mathematics, Pierce Expansion.

Cf. A000032, A006276, A001566, A002814, A045529, A112845, A219160, A219161, A219162.

NestList[#(#^2-3)&,3,6] (* Harvey P. Dale, Jun 09 2011 *)
RecurrenceTable[{a[n] == a[n - 1]^3 - 3*a[n - 1], a[0] == 3}, a, {n,
  0, 5}] (* G. C. Greubel, Dec 30 2016 *)

a(n)=2*fibonacci(2*3^n+1)-fibonacci(2*3^n)

a(n) = 2*F(2*3^n+1) - F(2*3^n) = ceiling(tau^(2*3^n)) where F(k) = A000045(k) is the k-th Fibonacci number and tau is the golden ratio. - Benoit Cloitre, Nov 29 2002
From Peter Bala, Nov 13 2012: (Start)
a(n) = ((3 + sqrt(5))/2)^(3^n) + ((3 - sqrt(5))/2)^(3^n).
Product_{n >= 0} (1 + 2/(a(n) - 1)) = sqrt(5).
a(n) = A002814(n+1) + 1. (End)
a(n) = 2*T(3^n,3/2), where T(n,x) denotes the n-th Chebyshev polynomial of the first kind. Cf. A219161. - Peter Bala, Feb 01 2017
From Amiram Eldar, Jan 12 2022: (Start)
a(n) = A000032(2*3^n).
a(n) = A006267(n)^2 + 2.
Product_{k=0..n} (a(k)-1) = Fibonacci(3^(n+1)) = A045529(n+1) (Janous, 2001). (End)
Sum_{n>=0} arctanh(1/a(n)) = log(5)/4 (Ohtsu, 2022). - Amiram Eldar, Dec 15 2022

cp's OEIS Frontend

A045521 Numbers k such that k! has initial digit '2'.

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A045522 Numbers k such that k! has initial digit '3'.

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A045523 Numbers k such that k! has initial digit '4'.

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

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A045525 Numbers k such that k! has initial digit '6'.

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A045526 Numbers k such that k! has initial digit '7'.

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A045527 Numbers k such that k! has initial digit '8'.

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