A265891 -id:A265891 - cp's OEIS frontend

A265897 Numbers n for which (2n+1)! / n! [= A000407(n)] is >= k! but < 2*k! for some k; positions of ones in A265891.

0, 1, 3, 8, 11, 14, 17, 20, 23, 30, 37, 41, 52, 56, 60, 64, 68, 72, 76, 89, 93, 97, 106, 110, 119, 128, 132, 137, 146, 155, 164, 169, 178, 183, 197, 202, 216, 221, 226, 231, 260, 265, 270, 275, 280, 285, 290, 295, 300, 331, 336, 341, 346, 351, 367, 372, 377, 393, 398, 403, 414, 419, 435, 440, 451, 456, 467, 472, 483, 494, 499, 510

Offset: 0

Antti Karttunen, Dec 24 2015

nonn

a(0) = 0 is a special case in this sequence, thus the indexing starts from zero.

Numbers n such that A000407(n) is in A265334.

Antti Karttunen, Table of n, a(n) for n = 0..845

Cf. A000407, A265891, A265334.

Cf. also A265898.

A099563 a(0) = 0; for n > 0, a(n) = final nonzero number in the sequence n, f(n,2), f(f(n,2),3), f(f(f(n,2),3),4),..., where f(n,d) = floor(n/d); the most significant digit in the factorial base representation of n.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 0

John W. Layman, Oct 22 2004

nonn

Records in {a(n)} occur at {1,4,18,96,600,4320,35280,322560,3265920,...}, which appears to be n*n! = A001563(n).

The most significant digit in the factorial expansion of n (A007623). Proof: The algorithm that computes the factorial expansion of n, generates the successive digits by repeatedly dividing the previous quotient with successively larger divisors (the remainders give the digits), starting from n itself and divisor 2. As a corollary we find that A001563 indeed gives the positions of the records. - Antti Karttunen, Jan 01 2007.

			For n=15, f(15,2) = floor(15/2)=7, f(7,3)=2, f(2,4)=0, so a(15)=2.
From _Antti Karttunen_, Dec 24 2015: (Start)
Example illustrating the role of this sequence in factorial base representation:
   n  A007623(n)       a(n) [= the most significant digit].
   0 =   0               0
   1 =   1               1
   2 =  10               1
   3 =  11               1
   4 =  20               2
   5 =  21               2
   6 = 100               1
   7 = 101               1
   8 = 110               1
   9 = 111               1
  10 = 120               1
  11 = 121               1
  12 = 200               2
  13 = 201               2
  14 = 210               2
  15 = 211               2
  16 = 220               2
  17 = 221               2
  18 = 300               3
  etc.
Note that there is no any upper bound for the size of digits in this representation.
(End)

Antti Karttunen, Table of n, a(n) for n = 0..10080
Index entries for sequences related to factorial base representation

Cf. A001563, A007623, A099564.

Cf. also A034968, A048764, A051683, A055881, A126307, A230420, A246359, A249069, A257679, A257684, A257686, A257687, A265890, A265891, A265894, A265333, A265334.

Table[Floor[n/#] &@ (k = 1; While[(k + 1)! <= n, k++]; k!), {n, 0, 120}] (* Michael De Vlieger, Aug 30 2016 *)

A099563(n) = { my(i=2,dig=0); until(0==n, dig = n % i; n = (n - dig)/i; i++); return(dig); }; \\ Antti Karttunen, Dec 24 2015

def a(n):
    i=2
    d=0
    while n:
        d=n%i
        n=(n - d)//i
        i+=1
    return d
print([a(n) for n in range(201)]) # Indranil Ghosh, Jun 21 2017, after PARI code

(define (A099563 n) (let loop ((n n) (i 2)) (let* ((dig (modulo n i)) (next-n (/ (- n dig) i))) (if (zero? next-n) dig (loop next-n (+ 1 i))))))
(definec (A099563 n) (cond ((zero? n) n) ((= 1 (A265333 n)) 1) (else (+ 1 (A099563 (A257684 n)))))) ;; Based on given recurrence, using the memoization-macro definec
;; Antti Karttunen, Dec 24-25 2015

From Antti Karttunen, Dec 25 2015: (Start)
a(0) = 0; for n >= 1, if A265333(n) = 1 [when n is one of the terms of A265334], a(n) = 1, otherwise 1 + a(A257684(n)).
Other identities. For all n >= 0:
a(A001563(n)) = n. [Sequence works as a left inverse for A001563.]
a(n) = A257686(n) / A048764(n).
(End)

a(0) = 0 prepended and the alternative description added to the name-field by Antti Karttunen, Dec 24 2015

A265894 a(n) = A099563(A001813(n)); the most significant digit in factorial base representation of (2n)! / n!.

Original entry on oeis.org

1, 1, 2, 1, 2, 6, 1, 4, 1, 2, 7, 1, 3, 10, 1, 3, 11, 1, 3, 10, 1, 3, 8, 25, 2, 6, 19, 1, 4, 13, 38, 2, 7, 23, 1, 4, 13, 39, 2, 6, 20, 1, 3, 9, 29, 1, 4, 13, 40, 1, 5, 16, 51, 2, 6, 20, 62, 2, 7, 23, 70, 2, 8, 25, 77, 2, 8, 25, 79, 2, 8, 25, 78, 2, 7, 23, 73, 2, 6, 21, 66, 1, 6, 18, 57, 1, 4, 15, 47, 1, 3, 12, 38, 118, 3, 9

Offset: 0

Antti Karttunen, Dec 24 2015

			The terms A001813(0) .. A001813(8) in factorial base representation (A007623) look as:
  1, 10, 200, 10000, 220000, 6000000, 174000000, 4760000000, 110000000000, ...
Taking the first digit (actually: a place holder value) of each gives the terms a(0) .. a(8) of this sequence: 1, 1, 2, 1, 2, 6, 1, 4, 1, ...

Antti Karttunen, Table of n, a(n) for n = 0..10080
Index entries for sequences related to factorial base representation

Submain diagonal of A265890.
Cf. A001813, A099563.
Cf. A265898 (positions of ones), A265899 (of descents), A266120 (local maxima just before those descents).
Cf. also A265891.

factBaseIntDs[n_] := Module[{m, i, len, dList, currDigit}, i = 1; While[n > i!, i++ ]; m = n; len = i; dList = Table[0, {len}]; Do[ currDigit = 0; While[m >= j!, m = m - j!; currDigit++ ]; dList[[len - j + 1]] = currDigit, {j, i, 1, -1}]; If[dList[[1]] == 0, dList = Drop[dList, 1]]; dList]; (* taken from A007623,  Alonso del Arte, May 03 2006 *) f[n_] := factBaseIntDs[(2 n)!/n!][[1]]; Array[f, 96, 0] (* Robert G. Wilson v, Dec 25 2015 *)

allocatemem((2^31)); \\ Enough?
A099563(n) = { my(i=2,dig=0); until(0==n, dig = n % i; n = (n - dig)/i; i++); return(dig); };
A265894 = n -> A099563((2*n)! / n!);

(define (A265894 n) (A099563 (A001813 n)))

(define (A265894 n) (A265890bi (+ 1 n) n)) ;; Code for A265890bi given in A265890.

a(n) = A099563(A001813(n)).

a(n) = A265890(n+1, n).

A265890 Array read by ascending antidiagonals: A(n,k) = A099563(A265609(n,k)), with n as row >= 0, k as column >= 0; the most significant digit in the factorial base representation of rising factorial n^(k) = (n+k-1)!/(n-1)!.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 2, 3, 2, 1, 1, 0, 1, 1, 1, 1, 3, 1, 1, 0, 1, 1, 1, 1, 1, 3, 1, 1, 0, 1, 1, 2, 2, 2, 1, 4, 1, 1, 0, 1, 1, 3, 4, 4, 3, 1, 4, 1, 1, 0, 1, 1, 3, 1, 1, 6, 3, 1, 5, 1, 1, 0, 1, 1, 4, 1, 1, 1, 8, 4, 1, 5, 1, 1, 0, 1, 2, 1, 1, 2, 2, 1, 1, 5, 2, 6, 1, 1, 0, 1, 2, 1, 2, 3, 3, 3, 2, 1, 6, 2, 6, 1, 1, 0

Offset: 0

Antti Karttunen, Dec 19 2015

Square array A(row,col) is read by ascending antidiagonals as: A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), A(3,0), A(2,1), A(1,2), A(0,3), ...

A265609(n,k) is the rising factorial, also known as Pochhammer symbol and A099563(n) is the most significant "digit" (place holder) in the factorial representation (A007623) of n.

			The top left corner of the array A265609 with its terms shown in factorial base (A007623) looks like this:
1,   0,    0,     0,       0,        0,         0,          0,           0
1,   1,   10,   100,    1000,    10000,    100000,    1000000,    10000000
1,  10,  100,  1000,   10000,   100000,   1000000,   10000000,   100000000
1,  11,  200,  2200,   30000,   330000,   4000000,   44000000,   500000000
1,  20,  310, 10000,  110000,  1220000,  14000000,  160000000,  1830000000
1,  21, 1100, 13300,  220000,  3000000,  36000000,  452000000,  5500000000
1, 100, 1300, 24000,  411000,  6000000,  82000000, 1100000000, 13300000000
1, 101, 2110, 41000, 1000000, 13000000, 174000000, 2374000000, 30360000000
-
Taking the most significant "digit" (placeholder that may get arbitrarily large values) gives us the top left corner of this array:
-
1, 0, 0, 0, 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0
1, 1, 1, 1, 1, 1, 1, 1,  1, 1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1
1, 1, 1, 1, 1, 1, 1, 1,  1, 1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1
1, 1, 2, 2, 3, 3, 4, 4,  5, 5,  6,  6,  7,  7,  8,  8,  9,  9, 10, 10, 11
1, 2, 3, 1, 1, 1, 1, 1,  1, 2,  2,  2,  2,  2,  2,  3,  3,  3,  3,  3,  3
1, 2, 1, 1, 2, 3, 3, 4,  5, 6,  7,  8, 10, 11, 12, 14, 15, 17, 19, 21,  1
1, 1, 1, 2, 4, 6, 8, 1,  1, 1,  1,  2,  2,  2,  2,  3,  3,  3,  4,  4,  5
1, 1, 2, 4, 1, 1, 1, 2,  3, 3,  4,  5,  6,  8,  9, 11, 12, 14, 16, 19, 21
1, 1, 3, 1, 1, 2, 3, 4,  6, 8, 11, 14,  1,  1,  1,  1,  2,  2,  2,  3,  3
1, 1, 3, 1, 2, 3, 5, 8,  1, 1,  1,  2,  2,  3,  4,  5,  6,  7,  8, 10, 12
1, 1, 4, 1, 3, 5, 9, 1,  2, 2,  3,  5,  6,  8, 11, 14, 17, 21,  1,  1,  1
1, 1, 1, 2, 4, 8, 1, 2,  3, 5,  7, 10, 14,  1,  1,  1,  2,  2,  3,  3,  4
1, 2, 1, 3, 6, 1, 2, 4,  6, 9, 14,  1,  1,  2,  3,  4,  5,  6,  8, 10, 13
1, 2, 1, 3, 1, 2, 3, 6, 10, 1,  1,  2,  3,  5,  6,  9, 12, 16, 21,  1,  1
1, 2, 1, 4, 1, 2, 5, 9,  1, 2,  3,  4,  7, 10, 14, 20,  1,  1,  2,  2,  3
1, 2, 2, 5, 1, 3, 7, 1,  2, 3,  5,  8, 13,  1,  1,  1,  2,  3,  4,  6,  8
...

Cf. A007623, A265609.
Column 1: A099563.
Row 0: A000007, rows 1 & 2: A000012, row 3: A008619 (see comment in A001710).
Row 4: 1,2,3 followed by A097992 ?
Main diagonal: A265891 (essentially, without the initial 1 from the corner of this array).
Cf. also array A265892.

(define (A265890 n) (A265890bi (A025581 n) (A002262 n)))
(define (A265890bi row col) (A099563 (A265609bi row col))) ;; Code for A265609bi given in A265609.

A216377 The most significant digit in base n representation of n!.

Original entry on oeis.org

1, 2, 1, 4, 3, 2, 1, 6, 3, 2, 1, 7, 4, 2, 1, 10, 5, 2, 1, 15, 8, 4, 2, 1, 13, 6, 3, 1, 25, 12, 6, 3, 1, 25, 12, 6, 3, 1, 28, 13, 6, 3, 1, 33, 16, 7, 3, 1, 41, 20, 9, 4, 2, 1, 26, 12, 6, 2, 1, 38, 18, 8, 3, 1, 57, 27, 12, 5, 2, 1, 43, 20, 9, 4, 2, 72, 33, 15, 7, 3, 1

Offset: 2

Alex Ratushnyak, Sep 06 2012

a(n) < n, by definition.
Numbers n such that a(n)=1: 2, 4, 8, 12, 16, 20, 25, 29, 34, 39, 44, 49, 55, 60, 65, 71, 82, 88, 94, 105, 111, 117, 123, 136, ... (see A221707).
Numbers n such that a(n) > a(k) for k < n: 2, 3, 5, 9, 13, 17, 21, 30, 40, 45, 50, 66, 77, 100, 118, 124, 130, 155, 161, 226, 246, 273, 371, 378, 385, 421, 450, 472, 509, 584, 599, 637, 660, 683, 745, 784, 855, 983, 991, 999, ... (see A221708).

Alois P. Heinz, Table of n, a(n) for n = 2..10000

Cf. A216021, A215894, A221707, A221708.

Cf. also to scatter plots of A265891 and A265894.

a:= n-> iquo(n!, n^ilog[n](n!)):
seq(a(n), n=2..100);  # Alois P. Heinz, Sep 06 2012

Table[IntegerDigits[n!, n][[1]], {n, 2, 100}] (* T. D. Noe, Sep 06 2012 *)

import math
def modlg(a,b):
    return a // b**int(math.log(a,b))
for n in range(2,88):
    print(modlg(math.factorial(n), n), end=', ')

a(n) = modlg(n!, n), where modlg is the function defined in A215894: modlg(A,B) = floor(A / B^floor(logB(A))), logB is the logarithm base B.

cp's OEIS Frontend

A265897 Numbers n for which (2n+1)! / n! [= A000407(n)] is >= k! but < 2*k! for some k; positions of ones in A265891.

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A099563 a(0) = 0; for n > 0, a(n) = final nonzero number in the sequence n, f(n,2), f(f(n,2),3), f(f(f(n,2),3),4),..., where f(n,d) = floor(n/d); the most significant digit in the factorial base representation of n.

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A265894 a(n) = A099563(A001813(n)); the most significant digit in factorial base representation of (2n)! / n!.

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A265890 Array read by ascending antidiagonals: A(n,k) = A099563(A265609(n,k)), with n as row >= 0, k as column >= 0; the most significant digit in the factorial base representation of rising factorial n^(k) = (n+k-1)!/(n-1)!.

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A216377 The most significant digit in base n representation of n!.

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