A265894 -id:A265894 - cp's OEIS frontend

A265898 Numbers n for which (2n)! / n! [= A001813(n)] is >= k! but < 2*k! for some k; positions of ones in A265894.

0, 1, 3, 6, 8, 11, 14, 17, 20, 27, 34, 41, 45, 49, 81, 85, 89, 102, 106, 115, 124, 128, 137, 142, 146, 151, 160, 169, 174, 188, 193, 202, 207, 212, 231, 236, 241, 246, 251, 256, 306, 311, 316, 321, 326, 331, 336, 357, 362, 367, 383, 388, 393, 409, 414, 425, 430, 446, 462, 478, 489, 494, 505, 516, 521, 532, 543, 554, 565

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 A001813(n) is in A265334.

Antti Karttunen, Table of n, a(n) for n = 0..843

After zero-term, a subsequence of A265899.
Cf. A001813, A265894, A265334.
Cf. also A265897.

A265899 After a(1) = 1, positions of descents in A265894.

Original entry on oeis.org

1, 3, 6, 8, 11, 14, 17, 20, 24, 27, 31, 34, 38, 41, 45, 49, 53, 57, 61, 65, 69, 73, 77, 81, 85, 89, 94, 98, 102, 106, 111, 115, 120, 124, 128, 133, 137, 142, 146, 151, 156, 160, 165, 169, 174, 179, 184, 188, 193, 198, 202, 207, 212, 217, 222, 227, 231, 236, 241, 246, 251, 256, 261, 266, 271, 276, 281, 286, 291, 296, 301

Offset: 1

Antti Karttunen, Dec 24 2015

nonn

Numbers n for which A099563(A001813(n)) <= A099563(A001813(n-1)), where A001813(n) = (2n)! / n!, and A099563 gives the most significant digit in the factorial base representation (A007623) of n.

Antti Karttunen, Table of n, a(n) for n = 1..1503

Cf. A001813, A007623, A099563, A265894.
Cf. A265898 (a subsequence), A266119 (first differences), A266120 (terms immediately before descents).
Cf. also A031435.

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!);
my(i=0, p=1, n=0); while(i < 60, n++; my(k = A265894(n)); if(k <= p, i++; print1(n, ", ")); p = k; );

;; With Antti Karttunen's IntSeq-library.
(define A265899 (MATCHING-POS 1 1 (lambda (n) (<= (A265894 n) (A265894 (- n 1))))))

A266120 Local maxima of A265894 just before its descents: a(n) = A265894(A265899(n) - 1).

Original entry on oeis.org

1, 2, 6, 4, 7, 10, 11, 10, 25, 19, 38, 23, 39, 20, 29, 40, 51, 62, 70, 77, 79, 78, 73, 66, 57, 47, 118, 91, 68, 49, 106, 71, 147, 93, 57, 108, 62, 112, 61, 105, 174, 89, 141, 68, 104, 154, 224, 100, 139, 191, 80, 105, 136, 172, 215, 263, 98, 116, 135, 154, 174, 192, 210, 225, 238, 248, 254, 257, 256, 251, 244, 233, 219, 204, 187, 169, 151, 133

Offset: 1

Antti Karttunen, Dec 24 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2016

Cf. A265894, A265899.

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!);
i=0; p=1; n=0; while(i < 2016, n++; k = A265894(n); if(k <= p, i++; write("b266120.txt", i, " ", p)); p = k; );

(define (A266120 n) (A265894 (- (A265899 n) 1)))

a(n) = A265894(A265899(n) - 1).

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

A265891 a(n) = A099563(A000407(n)); the most significant digit in factorial base representation of (2n+1)! / n!.

Original entry on oeis.org

1, 1, 2, 1, 3, 8, 2, 6, 1, 3, 10, 1, 5, 14, 1, 5, 16, 1, 5, 15, 1, 4, 12, 1, 3, 9, 28, 2, 6, 19, 1, 3, 11, 35, 2, 6, 19, 1, 3, 10, 30, 1, 4, 14, 44, 2, 6, 20, 61, 2, 8, 25, 1, 3, 10, 31, 1, 3, 11, 35, 1, 4, 12, 38, 1, 4, 12, 39, 1, 4, 12, 39, 1, 3, 11, 36, 1, 3, 10, 33, 102, 3, 9, 28, 89, 2, 7, 23, 74, 1, 6, 19, 59

Offset: 0

Antti Karttunen, Dec 20 2015

			The terms A000407(0) .. A000407(8) in factorial base representation (A007623) look as:
  1, 100, 2200, 110000, 3000000, 82000000, 2374000000, 65500000000, 1550000000000, ...
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, 3, 8, 2, 6, 1, ...

Antti Karttunen, Table of n, a(n) for n = 0..10080
Antti Karttunen, Scatter-plot of the graph drawn for the terms n=0 .. 1024, with the help of OEIS-plot utility.
Index entries for sequences related to factorial base representation.

Cf. A000407, A007623, A099563.
Main diagonal of A265890 (apart from the corner term).
Cf. A265897 (positions of ones).
Cf. also A265894.

a[n_] := Module[{k = (2*n+1)!/n!, m = 2, r, d=0}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, If[r > 0, d = r]; m++]; d]; Array[a, 100, 0] (* Amiram Eldar, Feb 14 2024 *)

allocatemem((2^31)); \\ Enough?
A099563(n) = { my(i=2,dig=0); until(0==n, dig = n % i; n = (n - dig)/i; i++); return(dig); };
A265891 = n -> A099563(((2*n)+1)! / n!);

(define (A265891 n) (A099563 (A000407 n)))

(define (A265891 n) (A265890bi (+ 1 n) (+ 1 n))) ;; Code for A265890bi given in A265890.

a(n) = A099563(A000407(n)).

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

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

A265898 Numbers n for which (2n)! / n! [= A001813(n)] is >= k! but < 2*k! for some k; positions of ones in A265894.

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A265899 After a(1) = 1, positions of descents in A265894.

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A266120 Local maxima of A265894 just before its descents: a(n) = A265894(A265899(n) - 1).

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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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A265891 a(n) = A099563(A000407(n)); the most significant digit in factorial base representation of (2n+1)! / n!.

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

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