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.
a004086 = read . reverse . show -- Reinhard Zumkeller, Apr 11 2011
|.&.": i.@- 1e5 NB. Stephen Makdisi, May 14 2018
read transforms; A004086 := digrev; #cf "Transforms" link at bottom of page A004086:=proc(n) local s,t; if n<10 then n else s:=irem(n,10,'t'); while t>9 do s:=s*10+irem(t,10,'t') od: s*10+t fi end; # M. F. Hasler, Jan 29 2012
Table[FromDigits[Reverse[IntegerDigits[n]]], {n, 0, 75}]
IntegerReverse[Range[0,80]](* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 13 2018 *)
dig(n) = {local(m=n,r=[]); while(m>0,r=concat(m%10,r);m=floor(m/10));r}
A004086(n) = {local(b,m,r);r=0;b=1;m=dig(n);for(i=1,matsize(m)[2],r=r+b*m[i];b=b*10);r} \\ Michael B. Porter, Oct 16 2009
A004086(n)=fromdigits(Vecrev(digits(n))) \\ M. F. Hasler, Nov 11 2010, updated May 11 2015, Sep 13 2019
def A004086(n): return int(str(n)[::-1]) # Chai Wah Wu, Aug 30 2014
11 in base 4 is 23: moving the most significant digit as the least significant one we have 32 that is 14 in base 10.
with(numtheory): P:=proc(q,h) local a,b,k,n; print(0); for n from 1 to q do a:=convert(n,base,h); b:=[]; for k from 1 to nops(a)-1 do b:=[op(b),a[k]]; od; a:=[a[nops(a)],op(b)]; a:=convert(a,base,h,10); b:=0; for k from nops(a) by -1 to 1 do b:=10*b+a[k]; od; print(b); od; end: P(10^4,4);
roll[n_, b_] := Block[{w = IntegerDigits[n, b]}, Append[Rest@ w, First@ w]]; b = 4; FromDigits[#, b] & /@ (roll[#, b] & /@ Range[0, 70]) (* Michael De Vlieger, Mar 04 2015 *)
Table[FromDigits[RotateLeft[IntegerDigits[n,4]],4],{n,0,70}] (* Harvey P. Dale, Aug 07 2015 *)
def A255689(n): x=A007090(n) return int (x[1:]+x[0],4) # Indranil Ghosh, Feb 08 2017
a035524 n = a035524_list !! n a035524_list = iterate a055948 1 -- Reinhard Zumkeller, Oct 10 2011
nxt4[n_]:=Module[{idn4=IntegerDigits[n,4]},FromDigits[idn4+ Reverse[idn4], 4]]; NestList[nxt4,1,40] (* Harvey P. Dale, May 02 2011 *)
def reversedigits(n, b=10): # reverse digits of n in base b
x, y = n, 0
while x >= b:
x, r = divmod(x, b)
y = b*y + r
return b*y + x
A035524_list, l = [1], 1
for _ in range(50):
l += reversedigits(l,4)
A035524_list.append(l)
a055948 n = n + a030103 n -- Reinhard Zumkeller, Oct 10 2011
Table[n+FromDigits[Reverse[IntegerDigits[n,4]],4],{n,0,70}] (* Harvey P. Dale, Nov 24 2021 *)
Each a(n) is obtained by concatenating the original base-4 expansion of n (which comes to the left hand, i.e., the most significant side) with its mirror-image (which comes to the right hand, i.e., the least significant side). For example, for a(4) we have 4 = '10' in base 4, which concatenated with its reversal '01' yields '1001', which when converted back to decimal yields 1*64 + 0*16 + 0*4 + 1*1 = 65, thus a(4)=65.
A048703(n) := (n) -> (2^(floor_log_2_coarse(n)+1))*n + sum('(bit_i(n, i+((-1)^i))*(2^(floor_log_2_coarse(n)-i)))', 'i'=0..floor_log_2_coarse(n)); bit_i := (x,i) -> `mod`(floor(x/(2^i)),2); # Following is like floor_log_2 but even results are incremented by one: floor_log_2_coarse := proc(n) local nn,i: nn := n; for i from -1 to n do if(0 = nn) then RETURN(i+(1-(i mod 2))); fi: nn := floor(nn/2); od: end:
q[n_] := EvenQ[IntegerLength[n, 4]] && PalindromeQ[IntegerDigits[n, 4]]; Select[Range[0, 2400, 5], q] (* Amiram Eldar, May 27 2024 *)
def A048703(n): s = bin(n-1)[2:] if len(s) % 2: s = '0'+s t = [s[i:i+2] for i in range(0,len(s),2)] return int(''.join(t+t[::-1]),2) # Chai Wah Wu, Feb 26 2021
11 in base 4 is 23: moving the least significant digit to the most significant one we have 32 that is 14 in base 10.
with(numtheory): P:=proc(q,h) local a,b,k,n; print(0); for n from 1 to q do a:=convert(n,base,h); b:=[]; for k from 2 to nops(a) do b:=[op(b),a[k]]; od; a:=[op(b),a[1]]; a:=convert(a,base,h,10); b:=0; for k from nops(a) by -1 to 1 do b:=10*b+a[k]; od; print(b); od; end: P(10^4,4);
roll[n_, b_] := Block[{w = IntegerDigits[n, b]}, Prepend[Most@ w, Last@ w]]; b = 4; FromDigits[#, b] & /@ (roll[#, b] & /@ Range[0, 69]) (* Michael De Vlieger, Mar 04 2015 *)
Array[FromDigits[RotateRight[IntegerDigits[#,4]],4]&,70,0] (* Harvey P. Dale, Mar 01 2016 *)
def A255589(n): x=str(A007090(n)) return int(x[-1]+x[:-1],4) # Indranil Ghosh, Feb 03 2017
9077 is a term as the number of divisors of 9077 = tau(9077) = 4, and this equals the number of divisors of R(9077) when written and then read as a base-j number, with 2 <= j <= 10. See the table below for k = 9077. . base | k_base | R(k_base) | R(k_base)_10 | tau(R(k_base)_10) ---------------------------------------------------------------------------------- 2 | 10001101110101 | 10101110110001 | 11185 | 4 3 | 110110012 | 210011011 | 15421 | 4 4 | 2031311 | 1131302 | 6002 | 4 5 | 242302 | 203242 | 6697 | 4 6 | 110005 | 500011 | 38887 | 4 7 | 35315 | 51353 | 12533 | 4 8 | 21565 | 56512 | 23882 | 4 9 | 13405 | 50431 | 33157 | 4 10 | 9077 | 7709 | 7709 | 4
Select[Range@100000,Length@Union@DivisorSigma[0,Join[{s=#},FromDigits[Reverse@IntegerDigits[s,#],#]&/@Range[2,10]]]==1&] (* Giorgos Kalogeropoulos, Jul 06 2021 *)
isok(k) = {my(t= numdiv(k)); for (b=2, 10, my(d=digits(k, b)); if (numdiv(fromdigits(Vecrev(d), b)) != t, return (0));); return(1);} \\ Michel Marcus, Jul 06 2021
The triangle starts 1; 3, 1; 1, 4, 1; 5, 7, 5, 1; 3, 2, 9, 6, 1;
A191780 := proc(n,k) d := ListTools[Reverse](convert(n,base,k)) ; add( op(i,d)*k^(i-1),i=1..nops(d)) ; end proc: # R. J. Mathar, Aug 26 2011
G[n_, k_] := IntegerDigits[n, k] // Reverse // FromDigits[#, k]&; Table[ G[n, k], {n, 2, 15}, {k, 2, n}] // Flatten (* Jean-François Alcover, Feb 10 2018 *)
For n = 6, the reversal of base 4 digits of n (written in base 10) is 9. So, a(6) = 6 - 9 = -3. - _Indranil Ghosh_, Feb 01 2017
Table[n-FromDigits[Reverse[IntegerDigits[n,4]],4],{n,0,90}] (* Harvey P. Dale, Aug 22 2011 *)
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