A048764 -id:A048764 - cp's OEIS frontend

A257686 a(0) = 0, for n >= 1: a(n) = A099563(n) * A048764(n).

0, 1, 2, 2, 4, 4, 6, 6, 6, 6, 6, 6, 12, 12, 12, 12, 12, 12, 18, 18, 18, 18, 18, 18, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 72

Offset: 0

Antti Karttunen, May 04 2015

For n >= 1, a(n) = the smallest term of A051683 >= n.
Can also be obtained by replacing with zeros all other digits except the first (the most significant) in the factorial base representation of n (A007623), then converting back to decimal.
Useful when computing A257687.

			Factorial base representation (A007623) of 2 is "10", zeroing all except the most significant digit does not change anything, thus a(2) = 2.
Factorial base representation (A007623) of 3 is "11", zeroing all except the most significant digit gives "10", thus a(3) = 2.
Factorial base representation of 23 is "321", zeroing all except the most significant digit gives "300" which is factorial base representation of 18, thus a(23) = 18.

Antti Karttunen, Table of n, a(n) for n = 0..5040

Cf. A007623, A048764, A051683, A099563, A257687.

Cf. also A053644 (analogous sequence for base-2).

from sympy import factorial as f
def a007623(n, p=2): return n if nIndranil Ghosh, Jun 21 2017

(define (A257686 n) (if (zero? n) n (* (A099563 n) (A048764 n))))

a(0) = 0, and for n >= 1: a(n) = A099563(n) * A048764(n).
Other identities:
For all n >= 0, a(n) = n - A257687(n).
a(n) = A000030(A007623(n))*(A055642(A007623(n)))! - Indranil Ghosh, Jun 21 2017

A276149 a(0) = 0; for n >= 1, a(n) = A048764(n) * (1+(A084558(n)-A099563(n))).

Original entry on oeis.org

0, 1, 4, 4, 2, 2, 18, 18, 18, 18, 18, 18, 12, 12, 12, 12, 12, 12, 6, 6, 6, 6, 6, 6, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 48

Offset: 0

Antti Karttunen, Aug 29 2016

Auxiliary function for computing A225901: the most significant digit in factorial base representation of n is "inverted", the rest of digits are "cleared" (replaced with zeros).

Antti Karttunen, Table of n, a(n) for n = 0..5039
Index entries for sequences related to factorial base representation

Cf. A048764, A084558, A099563, A225901.

{0}~Join~Table[# (1 + ((m = 1; While[m! <= n, m++]; m - 1) - Floor[n/#])) &[k = 1; While[(k + 1)! <= n, k++]; k!], {n, 96}] (* Michael De Vlieger, Aug 30 2016, after Jayanta Basu at A084558 *)

(define (A276149 n) (if (zero? n) n (* (A048764 n) (+ 1 (- (A084558 n) (A099563 n))))))

a(0) = 0; for n >= 1, a(n) = A048764(n) * (1+(A084558(n)-A099563(n))).

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

A260188 Greatest primorial less than or equal to n.

Original entry on oeis.org

1, 2, 2, 2, 2, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30, 30

Offset: 1

Jean-Marc Rebert, Jul 18 2015

nonn

			a(5) = 2 because 2 is the greatest primorial less than or equal to 5.
a(31) = 30 because 30 is the greatest primorial less than or equal to 31.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A034386 (primorials), A048764, A249270.

Table[k = 0; While[Times @@ Prime@ Range[k + 1] <= n, k++]; Times @@ Prime@ Range@ k, {n, 120}] (* Michael De Vlieger, Aug 30 2016 *)

a(n)=my(t=1,k); forprime(p=2,, k=t*p; if(k>n, return(t), t=k)) \\ Charles R Greathouse IV, Jul 20 2015

a(n) = max_{A034386(i) <= n} A034386(i).
a(n) >> n/log n. - Charles R Greathouse IV, Jul 20 2015
Sum_{n>=1} 1/a(n)^2 = A249270. - Amiram Eldar, Aug 09 2022

A278225 Filter-sequence for factorial base (cycles in A060117/A060118-permutations): Least number with the same prime signature as A275725.

Original entry on oeis.org

2, 4, 12, 8, 12, 8, 60, 36, 24, 16, 24, 16, 60, 24, 24, 16, 36, 16, 60, 24, 36, 16, 24, 16, 420, 180, 180, 72, 180, 72, 120, 72, 48, 32, 48, 32, 120, 48, 48, 32, 72, 32, 120, 48, 72, 32, 48, 32, 420, 180, 120, 48, 120, 48, 120, 72, 48, 32, 48, 32, 180, 72, 48, 32, 72, 32, 180, 72, 72, 32, 48, 32, 420, 120, 120, 48, 180, 48, 180, 72, 48, 32, 72, 32, 120, 48, 48

Offset: 0

Antti Karttunen, Nov 16 2016

nonn

This sequence can be used for filtering certain sequences related to cycle-structures in finite permutations as ordered by lists A060117 / A060118 (and thus also related to factorial base representation, A007623) because it matches only with any such sequence b that can be computed as b(n) = f(A275725(n)), where f(n) is any function that depends only on the prime signature of n (some of these are listed under the index entry for "sequences computed from exponents in ...").

Matching in this context means that the sequence a matches with the sequence b iff for all i, j: a(i) = a(j) => b(i) = b(j). In other words, iff the sequence b partitions the natural numbers to the same or coarser equivalence classes (as/than the sequence a) by the distinct values it obtains.

Cf. A007623, A060117, A060118, A046523, A275725.
Other filter-sequences related to factorial base: A278234, A278235, A278236.
Sequences that partition N into same or coarser equivalence classes: A048764, A048765, A060129, A060130, A060131, A084558, A275803, A275851, A257510.

(define (A278225 n) (A046523 (A275725 n)))

a(n) = A046523(A275725(n)).

A048765 Smallest factorial >= n.

Original entry on oeis.org

1, 2, 6, 6, 6, 6, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120

Offset: 1

Charles T. Le (charlestle(AT)yahoo.com)

J. Castillo, Other Smarandache Type Functions: Inferior/Superior Smarandache f-part of x, Smarandache Notions Journal, Vol. 10, No. 1-2-3 (1999), 202-204.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Krassimir T. Atanassov, On Some of Smarandache's Problems.
Vassia K. Atanassova and Krassimir T. Atanassov, On the 43rd and 44th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5, No. 2, (1999), 86-88.
Florentin Smarandache, Only Problems, Not Solutions!.

Cf. A000142, A001113, A001620, A048764, A229837.

a048764 n = a048764_list !! (n-1)
a048764_list = f [1..] $ tail a000142_list where
   f (u:us) vs'@(v:vs) | u == v    = v : f us vs
                       | otherwise = v : f us vs'
-- Reinhard Zumkeller, Jun 04 2012

Join[{1},Flatten[Table[Table[n!,n!-(n-1)!],{n,5}]]] (* Harvey P. Dale, Jun 15 2016 *)

a(n)=my(t=1,k=1);while(tCharles R Greathouse IV, Sep 19 2012

n <= a(n) << n log n / log log n. - Charles R Greathouse IV, Sep 19 2012

Sum_{n>=1} 1/a(n)^2 = 1 + Sum_{n>=1} (n!-(n-1)!)/n!^2 = e + gamma - Ei(1) = A001113 - A229837 = 1.4003796770..., where gamma is Euler's constant (A001620) and Ei is the exponential integral. - Amiram Eldar, Aug 09 2022

A135996 Difference between 2^n and the largest factorial <= 2^n.

Original entry on oeis.org

0, 0, 2, 2, 10, 8, 40, 8, 136, 392, 304, 1328, 3376, 3152, 11344, 27728, 25216, 90752, 221824, 161408, 685696, 1734272, 565504, 4759808, 13148416, 29925632, 27192064, 94300928, 228518656, 57869312, 594740224, 1668482048, 3815965696, 2362913792

Offset: 0

Ctibor O. Zizka, Mar 03 2008, Mar 16 2008

			a(6) = 2^6 - 4! = 40
a(7) = 2^7 - 5! = 8
a(8) = 2^8 - 5! = 136

Cf. A000142.

A048764 := proc(n) local a; for a from 1 do if a! > n then RETURN((a-1)!); fi ; od: end: A135996 := proc(n) 2^n-A048764(2^n) ; end: seq(A135996(n),n=0..60) ; # R. J. Mathar, Mar 16 2008

d2n[n_]:=Module[{n2=2^n,fcts=Reverse[Range[100]!]},n2-Select[fcts,#<= n2&,1]]; Flatten[Array[d2n,40,0]] (* Harvey P. Dale, Aug 22 2012 *)

a(n) = 2^n - A048764(2^n). - R. J. Mathar, Mar 16 2008

More terms from R. J. Mathar, Mar 16 2008

cp's OEIS Frontend

A257686 a(0) = 0, for n >= 1: a(n) = A099563(n) * A048764(n).

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A276149 a(0) = 0; for n >= 1, a(n) = A048764(n) * (1+(A084558(n)-A099563(n))).

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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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A260188 Greatest primorial less than or equal to n.

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A278225 Filter-sequence for factorial base (cycles in A060117/A060118-permutations): Least number with the same prime signature as A275725.

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A048765 Smallest factorial >= n.

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A135996 Difference between 2^n and the largest factorial <= 2^n.

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