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.
b = MixedRadix[Reverse@ Range[2, 12]]; h[n_] := Module[{s = 0, i = 2, k = n}, While[k > 0, k = Floor[n/i!]; s = s + (i - 1) k; i++]; n - s]; Table[h@ FromDigits[Map[Boole[# > 0] &, #] (Reverse@ Range[2, Length@ # + 1] - #), b] &@ IntegerDigits[n, b], {n, 0, 120}] (* 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]}]; h[n_] := Module[{s = 0, i = 2, k = n}, While[k > 0, k = Floor[n/i!]; s = s + (i - 1) k; i++]; n - s]; Table[h@ g[Map[Boole[# > 0] &, #] (Reverse@ Range[2, Length@ # + 1] - #)] &@ f@ n, {n, 0, 120}] (* Michael De Vlieger, Aug 29 2016, function h after Jean-François Alcover at A034968 *)
(define (A276146 n) (A034968 (A225901 n)))
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1])); A003557(n) = (n/factorback(factorint(n)[, 1])); A276076(n) = { my(i=0,m=1,f=1,nextf); while((n>0),i=i+1; nextf = (i+1)*f; if((n%nextf),m*=(prime(i)^((n%nextf)/f));n-=(n%nextf));f=nextf); m; }; A351952(n) = { my(u=A276076(n)); (A003415(u) / A003557(u)); }; A225901(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 A225901. A351953(n) = A351952(A225901(n));
a(205) = a(1!*1 + 3!*2 + 4!*3 + 5!*1) = 1+2+3+1 = 7. [corrected by Shin-Fu Tsai, Mar 23 2021] From _Joerg Arndt_, Jun 17 2011: (Start) n: permutation inv. table a(n) cycles 0: [ 0 1 2 3 ] [ 0 0 0 ] 0 (0) (1) (2) (3) 1: [ 0 1 3 2 ] [ 0 0 1 ] 1 (0) (1) (2, 3) 2: [ 0 2 1 3 ] [ 0 1 0 ] 1 (0) (1, 2) (3) 3: [ 0 2 3 1 ] [ 0 1 1 ] 2 (0) (1, 2, 3) 4: [ 0 3 1 2 ] [ 0 2 0 ] 2 (0) (1, 3, 2) 5: [ 0 3 2 1 ] [ 0 2 1 ] 3 (0) (1, 3) (2) 6: [ 1 0 2 3 ] [ 1 0 0 ] 1 (0, 1) (2) (3) 7: [ 1 0 3 2 ] [ 1 0 1 ] 2 (0, 1) (2, 3) 8: [ 1 2 0 3 ] [ 1 1 0 ] 2 (0, 1, 2) (3) 9: [ 1 2 3 0 ] [ 1 1 1 ] 3 (0, 1, 2, 3) 10: [ 1 3 0 2 ] [ 1 2 0 ] 3 (0, 1, 3, 2) 11: [ 1 3 2 0 ] [ 1 2 1 ] 4 (0, 1, 3) (2) 12: [ 2 0 1 3 ] [ 2 0 0 ] 2 (0, 2, 1) (3) 13: [ 2 0 3 1 ] [ 2 0 1 ] 3 (0, 2, 3, 1) 14: [ 2 1 0 3 ] [ 2 1 0 ] 3 (0, 2) (1) (3) 15: [ 2 1 3 0 ] [ 2 1 1 ] 4 (0, 2, 3) (1) 16: [ 2 3 0 1 ] [ 2 2 0 ] 4 (0, 2) (1, 3) 17: [ 2 3 1 0 ] [ 2 2 1 ] 5 (0, 2, 1, 3) 18: [ 3 0 1 2 ] [ 3 0 0 ] 3 (0, 3, 2, 1) 19: [ 3 0 2 1 ] [ 3 0 1 ] 4 (0, 3, 1) (2) 20: [ 3 1 0 2 ] [ 3 1 0 ] 4 (0, 3, 2) (1) 21: [ 3 1 2 0 ] [ 3 1 1 ] 5 (0, 3) (1) (2) 22: [ 3 2 0 1 ] [ 3 2 0 ] 5 (0, 3, 1, 2) 23: [ 3 2 1 0 ] [ 3 2 1 ] 6 (0, 3) (1, 2) (End)
[seq(convert(fac_base(j),`+`),j=0..119)]; # fac_base and PermRevLexUnrank given in A055089. Perm2InversionVector in A064039 Or alternatively: [seq(convert(Perm2InversionVector(PermRevLexUnrank(j)),`+`),j=0..119)]; # third Maple program: b:= proc(n, i) local q; `if`(n=0, 0, b(irem(n, i!, 'q'), i-1)+q) end: a:= proc(n) local k; for k while k!Alois P. Heinz, Nov 15 2012
a[n_] := Module[{s=0, i=2, k=n}, While[k > 0, k = Floor[n/i!]; s = s + (i-1)*k; i++]; n-s]; Table[a[n], {n, 0, 105}] (* Jean-François Alcover, Nov 06 2013, after Benoit Cloitre *)
a(n)=local(k,r);k=2;r=0;while(n>0,r+=n%k;n\=k;k++);r \\ Franklin T. Adams-Watters, May 13 2009
def a(n):
k=2
r=0
while n>0:
r+=n%k
n=n//k
k+=1
return r
print([a(n) for n in range(201)]) # Indranil Ghosh, Jun 19 2017, after PARI program
def A034968(n, p=2): return n if nA034968(n//p, p+1) + n%p print([A034968(n) for n in range(106)]) # Michael S. Branicky, Oct 27 2024
(define (A034968 n) (let loop ((n n) (i 2) (s 0)) (cond ((zero? n) s) (else (loop (quotient n i) (+ 1 i) (+ s (remainder n i))))))) ;; Antti Karttunen, Aug 29 2016
a004488 0 = 0
a004488 n = if d == 0 then 3 * a004488 n' else 3 * a004488 n' + 3 - d
where (n', d) = divMod n 3
-- Reinhard Zumkeller, Mar 12 2014
a:= proc(n) local t, r, i;
t, r:= n, 0;
for i from 0 while t>0 do
r:= r+3^i *irem(2*irem(t, 3, 't'), 3)
od; r
end:
seq(a(n), n=0..80); # Alois P. Heinz, Sep 07 2011
a[n_] := FromDigits[Mod[3-IntegerDigits[n, 3], 3], 3]; Table[a[n], {n, 0, 66}] (* Jean-François Alcover, Mar 03 2014 *)
a(n) = my(b=3); fromdigits(apply(d->(b-d)%b, digits(n, b)), b); vector(67, i, a(i-1)) \\ Gheorghe Coserea, Apr 23 2018
from sympy.ntheory.factor_ import digits
def a(n): return int("".join([str((3 - i)%3) for i in digits(n, 3)[1:]]), 3) # Indranil Ghosh, Jun 06 2017
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