This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
For n=19, A007623(19) = 301, thus a(19) = 5 because A007088(5) = 101.
a(1000) = a(1*6! + 2*5! + 1*4! + 2*3! + 2*2!) = (7-1)*6! + (6-2)*5! + (5-1)*4! + (4-2)*3! + (3-2)*2! = 4910. a(1397) = a(1*6! + 5*5! + 3*4! + 0*3! + 2*2! + 1*1!) = (7-1)*6! + (6-5)*5! + (5-3)*4! + (3-2)*2! + (2-1)*1! = 4491.
b = MixedRadix[Reverse@ Range[2, 12]]; Table[FromDigits[Map[Boole[# > 0] &, #] (Reverse@ Range[2, Length@ # + 1] - #), b] &@ IntegerDigits[n, b], {n, 0, 82}] (* Version 10.2, or *)
f[n_] := Block[{a = {{0, n}}}, Do[AppendTo[a, {First@ #, Last@ #} &@ QuotientRemainder[a[[-1, -1]], Times @@ Range[# - i]]], {i, 0, #}] &@ NestWhile[# + 1 &, 0, Times @@ Range[# + 1] <= n &]; Most@ Rest[a][[All, 1]] /. {} -> {0}]; g[w_List] := Total[Times @@@ Transpose@ {Map[Times @@ # &, Range@ Range[0, Length@ w]], Reverse@ Append[w, 0]}]; Table[g[Map[Boole[# > 0] &, #] (Reverse@ Range[2, Length@ # + 1] - #)] &@ f@ n, {n, 0, 82}] (* Michael De Vlieger, Aug 29 2016 *)
a(n)=my(s=0,d,k=2);while(n,d=n%k;n=n\k;if(d,s=s+(k-d)*(k-1)!);k=k+1);return(s)
from sympy import factorial as f
def a(n):
s=0
k=2
while(n):
d=n%k
n=(n//k)
if d: s=s+(k - d)*f(k - 1)
k+=1
return s
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 19 2017
(define (A225901 n) (let loop ((n n) (z 0) (m 2) (f 1)) (cond ((zero? n) z) (else (loop (quotient n m) (if (zero? (modulo n m)) z (+ z (* f (- m (modulo n m))))) (+ 1 m) (* f m)))))) ;; One implementing the first recurrence, with memoization-macro definec: (definec (A225901 n) (if (zero? n) n (+ (A276091 (A275736 n)) (A153880 (A225901 (A257684 n)))))) ;; Antti Karttunen, Aug 29 2016
128 is in the sequence since 5! + 3! + 2! = 128. a(22) = 128. a(22) = a(6) + (1 + floor(log(16) / log(2)))! = 8 + 5! = 128. Also, 22 = 10110_2. Therefore, a(22) = 1 * 5! + 0 * 4! + 1 * 3! + 1 + 2! + 0 * 0! = 128. - _David A. Corneth_, Aug 21 2016
import Data.List (elemIndices) a059590 n = a059590_list !! n a059590_list = elemIndices 1 $ map a115944 [0..] -- Reinhard Zumkeller, Dec 04 2011
[seq(bin2facbase(j),j=0..64)]; bin2facbase := proc(n) local i; add((floor(n/(2^i)) mod 2)*((i+1)!),i=0..floor_log_2(n)); end; floor_log_2 := proc(n) local nn,i; nn := n; for i from -1 to n do if(0 = nn) then RETURN(i); fi; nn := floor(nn/2); od; end; # next Maple program: a:= n-> (l-> add(l[j]*j!, j=1..nops(l)))(Bits[Split](n)): seq(a(n), n=0..57); # Alois P. Heinz, Aug 12 2025
a[n_] := Reverse[id = IntegerDigits[n, 2]].Range[Length[id]]!; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Jun 19 2012, after Philippe Deléham *)
a(n) = if(n>0, a(n-msb(n)) + (1+logint(n,2))!, 0)
msb(n) = 2^#binary(n)>>1
{my(b = binary(n)); sum(i=1,#b,b[i]*(#b+1-i)!)} \\ David A. Corneth, Aug 21 2016
def facbase(k, f):
return sum(f[i] for i, bi in enumerate(bin(k)[2:][::-1]) if bi == "1")
def auptoN(N): # terms up to N factorial-base digits; 13 generates b-file
f = [factorial(i) for i in range(1, N+1)]
return list(facbase(k, f) for k in range(2**N))
print(auptoN(5)) # Michael S. Branicky, Oct 15 2022
22 has factorial base representation "320" (= A007623(22)), which does not contain any "1". Thus a(22) = 1, as the empty product is 1. 35 has factorial base representation "1121" (= A007623(35)). 1's occur in the following positions, when counted from right, starting with 1: 1, 3 and 4. Thus a(35) = prime(1)*prime(3)*prime(4) = 2*5*7 = 70.
nn = 105; m = 1; While[Factorial@ m < nn, m++]; m; Map[Times @@ Map[Prime, Flatten@ Position[#, 1]] &@ Reverse@ IntegerDigits[#, MixedRadix[Reverse@ Range[2, m]]] &, Range[0, nn]] (* Michael De Vlieger, Aug 11 2016, Version 10.2 *)
from operator import mul from sympy import prime def a007623(n, p=2): return n if n
;; Recursive definition using memoizing definec-macro: (definec (A275732 n) (cond ((zero? (A257261 n)) 1) (else (* (A000040 (A257261 n)) (A275732 (A275730bi n (- (A257261 n) 1))))))) (define (A275732 n) (let loop ((z 1) (n n)) (let ((y (A257261 n))) (cond ((zero? y) z) (else (loop (* z (A000040 y)) (A275730bi n (- y 1)))))))) ;; Code for A275730bi given in A275730.
Columns 0-4 of rows 0 - 24 of the array:
0, 0, 0, 0, 0, ... [No matter which digit of zero we clear, it stays zero forever]
0, 1, 1, 1, 1 ... [When clearing the least significant digit (pos. 0) of one, "1", we get zero, and clearing any other digit past the most significant digit keeps one as one]
2, 0, 2, 2, 2, ... [Clearing the least significant digit of 2, "10", doesn't affect it, but clearing the digit-1 zeros the whole number].
2, 1, 3, 3, 3, ... [Clearing the least significant factorial base digit of 3 ("11") gives "10", 2, clearing the digit-1 gives "01" = 1, and clearing any digit past the most significant keeps "11" as it is, 3].
4, 0, 4, 4, 4
4, 1, 5, 5, 5
6, 6, 0, 6, 6
6, 7, 1, 7, 7
8, 6, 2, 8, 8
8, 7, 3, 9, 9
10, 6, 4, 10, 10
10, 7, 5, 11, 11
12, 12, 0, 12, 12
12, 13, 1, 13, 13
14, 12, 2, 14, 14
14, 13, 3, 15, 15
16, 12, 4, 16, 16
16, 13, 5, 17, 17
18, 18, 0, 18, 18
18, 19, 1, 19, 19
20, 18, 2, 20, 20
20, 19, 3, 21, 21
22, 18, 4, 22, 22
22, 19, 5, 23, 23
24, 24, 24, 0, 24
...
(define (A275730 n) (A275730bi (A002262 n) (A025581 n))) (define (A275730bi n c) (let loop ((z 0) (n n) (m 2) (f 1) (c c)) (let ((d (modulo n m))) (cond ((zero? n) z) ((zero? c) (loop z (/ (- n d) m) (+ 1 m) (* f m) (- c 1))) (else (loop (+ z (* f d)) (/ (- n d) m) (+ 1 m) (* f m) (- c 1)))))))
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