A053737 -id:A053737 - cp's OEIS frontend

A033095 Number of 1's when n is written in base b for 2<=b<=n+1.

1, 1, 3, 4, 6, 6, 9, 6, 10, 10, 12, 11, 16, 13, 15, 14, 16, 13, 18, 15, 21, 20, 21, 16, 24, 20, 23, 23, 26, 25, 32, 22, 26, 25, 25, 28, 34, 28, 32, 30, 35, 30, 37, 31, 35, 36, 35, 31, 41, 34, 37, 36, 39, 35, 43, 38, 44, 41, 42, 38, 49, 40, 43

Offset: 1

Clark Kimberling

Cf. A053735, A053737, A062756, A077267.

Cf. A033093, A033096, A033097, A033099, A033101, A033103, A033105, A033107, A033109, A033111.

f[n_] := Count[Flatten@ Table[ IntegerDigits[n, b], {b, 2, n + 1}], 1]; Array[f, 63] (* Robert G. Wilson v, Nov 14 2012 *)

G.f.: x+(Sum_{b>=2} (Sum_{k>=0} x^(b^k)/(Sum_{0<=iFranklin T. Adams-Watters, Nov 03 2005

A053839 a(n) = (sum of digits of n written in base 4) modulo 4.

Original entry on oeis.org

0, 1, 2, 3, 1, 2, 3, 0, 2, 3, 0, 1, 3, 0, 1, 2, 1, 2, 3, 0, 2, 3, 0, 1, 3, 0, 1, 2, 0, 1, 2, 3, 2, 3, 0, 1, 3, 0, 1, 2, 0, 1, 2, 3, 1, 2, 3, 0, 3, 0, 1, 2, 0, 1, 2, 3, 1, 2, 3, 0, 2, 3, 0, 1, 1, 2, 3, 0, 2, 3, 0, 1, 3, 0, 1, 2, 0, 1, 2, 3, 2, 3, 0, 1, 3, 0, 1, 2, 0, 1, 2, 3, 1, 2, 3, 0, 3, 0, 1, 2, 0, 1, 2, 3, 1

Offset: 0

Henry Bottomley, Mar 28 2000

a(n) is the third row of the array in A141803. - Andrey Zabolotskiy, May 16 2016

This is the fixed point of the morphism 0->0123, 1->1230, 2->2301, 3->3012 starting with 0. Let t be the (nonperiodic) sequence of positions of 0, and likewise, u for 1, v for 2, and w for 3; then t(n)/n -> 4, u(n)/n -> 4, v(n)/n -> 4, w(n)/n -> 4, and t(n) + u(n) + v(n) + w(n) = 16*n - 6 for n >= 1. - Clark Kimberling, May 31 2017

			First three iterations of the morphism 0->0123, 1->1230, 2->2301, 3->3012:
  0123
  0123123023013012
  0123123023013012123023013012012323013012012312303012012312302301

Robert Israel, Table of n, a(n) for n = 0..10000
Glen Joyce C. Dulatre, Jamilah V. Alarcon, Vhenedict M. Florida and Daisy Ann A. Disu, On Fractal Sequences, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 109-113.
Robert Walker, Self Similar Sloth Canon Number Sequences
Index entries for sequences that are fixed points of mappings

Cf. A010060, A053837-A053844, A141803, A287552, A287553, A287554, A287555.

Cf. A010873, A053737.

seq(convert(convert(n,base,4),`+`) mod 4, n=0..100); # Robert Israel, May 18 2016

Mod[Total@ IntegerDigits[#, 4], 4] & /@ Range[0, 120] (* Michael De Vlieger, May 17 2016 *)
s = Nest[Flatten[# /. {0 -> {0, 1, 2, 3}, 1 -> {1, 2, 3, 0}, 2 -> {2, 3, 0, 1}, 3 -> {3, 0, 1, 2}}] &, {0}, 9];   (* - Clark Kimberling, May 31 2017 *)

a(n) = vecsum(digits(n,4)) % 4; \\ Michel Marcus, May 16 2016

a(n) = sumdigits(n, 4) % 4; \\ Michel Marcus, Jul 04 2018

a(n) = A010873(A053737(n)). - Andrey Zabolotskiy, May 18 2016

G.f. G(x) satisfies x^81*G(x) - (x^72+x^75+x^78+x^81)*G(x^4) + (x^48+x^60+x^63-x^64+x^72+x^75-x^76+x^78-x^79-x^88-x^91-x^94)*G(x^16) + (-1+x^16-x^48-x^60-x^63+2*x^64+x^76+x^79-x^80+x^112+x^124+x^127-x^128-x^140-x^143)*G(x^64) + (1-x^16-x^64+x^80-x^256+x^272+x^320-x^336)*G(x^256) = 0. - Robert Israel, May 18 2016

A053831 Sum of digits of n written in base 11.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 7, 8, 9, 10, 11, 12, 13, 14, 15

Offset: 0

Henry Bottomley, Mar 28 2000

Also the fixed point of the morphism 0->{0,1,2,3,4,5,6,7,8,9,10}, 1->{1,2,3,4,5,6,7,8,9,10,11}, 2->{2,3,4,5,6,7,8,9,10,11,12}, etc. - Robert G. Wilson v, Jul 27 2006

			a(20) = 1 + 9 = 10 because 20 is written as 19 base 11.

Tanar Ulric, Table of n, a(n) for n = 0..10000
Jeffrey O. Shallit, Problem 6450, Advanced Problems, The American Mathematical Monthly, Vol. 91, No. 1 (1984), pp. 59-60; Two series, solution to Problem 6450, ibid., Vol. 92, No. 7 (1985), pp. 513-514.
Robert Walker, Self Similar Sloth Canon Number Sequences.
Eric Weisstein's World of Mathematics, Digit Sum.

Cf. A007953, A064458, A138530.

Sum of digits of n written in bases 2-16: A000120, A053735, A053737, A053824, A053827, A053828, A053829, A053830, A007953, this sequence, A053832, A053833, A053834, A053835, A053836.

int Base11DigitSum(int n) {
   int count = 0;
   while (n != 0) { count += n % 11; n = n / 11; }
   return count;
} // Tanar Ulric, Oct 20 2021

Table[Plus @@ IntegerDigits[n, 11], {n, 0, 86}] (* or *)
Nest[ Flatten[ #1 /. a_Integer -> Table[a + i, {i, 0, 10}]] &, {0}, 2] (* Robert G. Wilson v, Jul 27 2006 *)

a(n)=if(n<1,0,if(n%11,a(n-1)+1,a(n/11)))

a(n)=sumdigits(n,11) \\ Charles R Greathouse IV, Oct 20 2021

From Benoit Cloitre, Dec 19 2002: (Start)
a(0)=0, a(11n+i) = a(n)+i for 0 <= i <= 10.
a(n) = n-(m-1)*(Sum_{k>0} floor(n/m^k)) = n-(m-1)*A064458(n). (End)
a(n) = A138530(n,11) for n > 10. - Reinhard Zumkeller, Mar 26 2008
Sum_{n>=1} a(n)/(n*(n+1)) = 11*log(11)/10 (Shallit, 1984). - Amiram Eldar, Jun 03 2021

A244041 Sum of digits of n written in fractional base 4/3.

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 5, 6, 5, 6, 7, 8, 6, 7, 8, 9, 6, 7, 8, 9, 9, 10, 11, 12, 8, 9, 10, 11, 10, 11, 12, 13, 8, 9, 10, 11, 11, 12, 13, 14, 12, 13, 14, 15, 9, 10, 11, 12, 11, 12, 13, 14, 14, 15, 16, 17, 14, 15, 16, 17, 10, 11, 12, 13, 11, 12, 13, 14, 14, 15, 16, 17

Offset: 0

Hailey R. Olafson, Jun 17 2014

The base 4/3 expansion is unique and thus the sum of digits function is well-defined.

			In base 4/3 the number 14 is represented by 3212 and so a(14) = 3 + 2 + 1 + 2 = 8.

G. C. Greubel, Table of n, a(n) for n = 0..1000
F. M. Dekking, The Thue-Morse Sequence in Base 3/2, J. Int. Seq., Vol. 26 (2023), Article 23.2.3.
Kevin Ryde, Plot for Upper/Lower Bound Factors (and LaTeX source).
Index entries for sequences related to fractional bases

Cf. A000120, A007953, A024631, A053737, A244040.

p:=4; q:=3; a[n_]:= a[n]= If[n==0, 0, a[q*Floor[n/p]] + Mod[n, p]]; Table[a[n], {n,0,75}] (* G. C. Greubel, Aug 20 2019 *)

a(n) = p=4; q=3; if(n==0,0, a(q*(n\p)) + (n%p));
vector(75, n, n--; a(n)) \\ G. C. Greubel, Aug 20 2019

def base43sum(n):
    L, i = [n], 1
    while L[i-1]>3:
        x=L[i-1]
        L[i-1]=x.mod(4)
        L.append(3*floor(x/4))
        i+=1
    return sum(L)
[base43sum(n) for n in [0..75]]

a(n) = A007953(A024631(n)). - Michel Marcus, Jun 17 2014
a(n) < 3 log(n)/log(4/3) < 11 log(n) for n > 1. Possibly the constant factor can be replaced by 7 or 8. - Charles R Greathouse IV, Sep 22 2022
Conjecture: a(n) >> log(n), hence a(n) ≍ log(n). - Charles R Greathouse IV, Nov 03 2022

A173525 a(n) = 1 + A053824(n-1), where A053824 = sum of digits in base 5.

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 3, 4, 5, 6, 7, 4, 5, 6, 7, 8, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 3, 4, 5, 6, 7, 4, 5, 6, 7, 8, 5, 6, 7, 8, 9, 6, 7, 8, 9, 10, 3, 4, 5, 6, 7, 4, 5, 6, 7, 8, 5, 6, 7, 8, 9, 6, 7, 8, 9, 10, 7, 8, 9, 10, 11, 4, 5, 6, 7, 8, 5, 6, 7, 8, 9, 6, 7, 8, 9, 10, 7, 8, 9, 10, 11, 8, 9, 10, 11, 12

Offset: 1

Omar E. Pol, Feb 20 2010

Also: a(n) = A053824(5^k+n-1) in the limit k->infinity, where k plays the role of a row index in A053824. (See the comment by M. F. Hasler for the proof.)
This means: if A053824 is regarded as a triangle then the rows converge to this sequence.
See conjecture in the entry A000120, and the case of base 2 in A063787.
From R. J. Mathar, Dec 09 2010: (Start)
In base b=5, A053824 starts counting up from 1 each time the index wraps around a power of b: A053824(b^k)=1.
Obvious recurrences are A053824(m*b^k+i) = m+A053824(i), 1 <= m < b-1, 0 <= i < b^(k-1).
So A053824 can be decomposed into a triangle T(k,n) = A053824(b^k+n-1), assuming that column indices start at n=1; row lengths are (b-1)*b^k.
There is a self-similarity in these sequences; a sawtooth structure of periodicity b is added algebraically on top of a sawtooth structure of periodicity b^2, on top of a periodicity b^3 etc. This leads to some "fake" finitely periodic substructures in the early parts of each row of T(.,.): often, but not always, a(n+b)=1+a(n). Often, but not always, a(n+b^2)=1+a(n) etc.
The common part of the rows T(.,.) grows with the power of b as shown in the recurrence above, and defines a(n) in the limit of large row indices k. (End)
The two definitions agree because the first 5^r terms in each row correspond to numbers 5^r, 5^r+1,...,5^r+(5^r-1), which are written in base 5 as a leading 1 plus the digits of 0,...,5^r-1. - M. F. Hasler, Dec 09 2010
From Omar E. Pol, Dec 10 2010: (Start)
In the scatter plots of these sequences, the basic structure is an element with b^2 points, where b is the associated base. (Scatter plots are created with the "graph" button of a sequence.) Sketches of these structures look as follows, the horizontal axis a squeezed version of the index n, b consecutive points packed vertically, and the vertical axis a(n):
........................................................
................................................ * .....
............................................... ** .....
..................................... * ...... *** .....
.................................... ** ..... **** .....
.......................... * ...... *** .... ***** .....
......................... ** ..... **** ... ****** .....
............... * ...... *** .... ***** ... ***** ......
.............. ** ..... **** .... **** .... **** .......
.... * ...... *** ..... *** ..... *** ..... *** ........
... ** ...... ** ...... ** ...... ** ...... ** .........
... * ....... * ....... * ....... * ....... * ..........
........................................................
... b=2 ..... b=3 ..... b=4 ..... b=5 ..... b=6 ........
. A000120 . A053735 . A053737 . A053824 . A053827 ......
. A063787 . A173523 . A173524 . A173525 . A173526 ......
........................................................
............................................. * ........
............................................ ** ........
........................... * ............. *** ........
.......................... ** ............ **** ........
........... *............ *** ........... ***** ........
.......... ** .......... **** .......... ****** ........
......... ***.......... ***** ......... ******* ........
........ **** ........ ****** ........ ******** ........
....... ***** ....... ******* ....... ********* ........
...... ****** ...... ******** ....... ******** .........
..... ******* ...... ******* ........ ******* ..........
..... ****** ....... ****** ......... ****** ...........
..... ***** ........ ***** .......... ***** ............
..... **** ......... **** ........... **** .............
..... *** .......... *** ............ *** ..............
..... ** ........... ** ............. ** ...............
..... * ............ * .............. * ................
........................................................
..... b=7 .......... b=8 ............ b=9 ..............
... A053828 ...... A053829 ........ A053830 ............
... A173527 ...... A173528 ........ A173529 ............(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..3126=5^5+1
Robert Walker, Self Similar Sloth Canon Number Sequences

Cf. A000120, A053824, A063787, A173523, A173524, A173526, A173527, A173528, A173529.

a173525 = (+ 1) . a053824 . (subtract 1) -- Reinhard Zumkeller, Jan 31 2014

A053825 := proc(n) add(d, d=convert(n,base,5)) ; end proc:
A173525 := proc(n) local b,k; b := 5 ; if n < b then n; else k := n/(b-1);   k := ceil(log(k)/log(b)) ; A053825(b^k+n-1) ; end if; end proc:
seq(A173525(n),n=1..100) ;

Total[IntegerDigits[#,5]]+1&/@Range[0,100] (* Harvey P. Dale, Jun 14 2015 *)

A173525(n)={ my(s=1); n--; until(!n\=5, s+=n%5); s } \\ M. F. Hasler, Dec 09 2010

A173525(n)={ my(s=1+(n=divrem(n-1,5))[2]); while((n=divrem(n[1],5))[1],s+=n[2]); s+n[2] } \\ M. F. Hasler, Dec 09 2010

a(n) = A053824(5^k + n - 1) where k >= ceiling(log_5(n/4)). - R. J. Mathar, Dec 09 2010

More terms from Vincenzo Librandi, Aug 02 2010

A346688 Replace 4^k with (-1)^k in base-4 expansion of n.

Original entry on oeis.org

0, 1, 2, 3, -1, 0, 1, 2, -2, -1, 0, 1, -3, -2, -1, 0, 1, 2, 3, 4, 0, 1, 2, 3, -1, 0, 1, 2, -2, -1, 0, 1, 2, 3, 4, 5, 1, 2, 3, 4, 0, 1, 2, 3, -1, 0, 1, 2, 3, 4, 5, 6, 2, 3, 4, 5, 1, 2, 3, 4, 0, 1, 2, 3, -1, 0, 1, 2, -2, -1, 0, 1, -3, -2, -1, 0, -4, -3, -2, -1, 0, 1, 2, 3, -1, 0, 1, 2, -2, -1, 0, 1, -3, -2, -1, 0, 1, 2, 3, 4, 0, 1, 2, 3, -1

Offset: 0

Ilya Gutkovskiy, Jul 29 2021

If n has base-4 expansion abc..xyz with least significant digit z, a(n) = z - y + x - w + ...

			54 = 312_4, 2 - 1 + 3 = 4, so a(54) = 4.

Cf. A007090, A053737, A055017, A065359, A065368, A346689, A346690, A346691.

nmax = 104; A[] = 0; Do[A[x] = x (1 + 2 x + 3 x^2)/(1 - x^4) - (1 + x + x^2 + x^3) A[x^4] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[n + 5 Sum[(-1)^k Floor[n/4^k], {k, 1, Floor[Log[4, n]]}], {n, 0, 104}]

from sympy.ntheory.digits import digits
def a(n):
    return sum(bi*(-1)**k for k, bi in enumerate(digits(n, 4)[1:][::-1]))
print([a(n) for n in range(105)]) # Michael S. Branicky, Jul 29 2021

G.f. A(x) satisfies: A(x) = x * (1 + 2*x + 3*x^2) / (1 - x^4) - (1 + x + x^2 + x^3) * A(x^4).

a(n) = n + 5 * Sum_{k>=1} (-1)^k * floor(n/4^k).

A054893 a(n) = Sum_{j > 0} floor(n/4^j).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 23, 23, 24, 24, 24, 24

Offset: 0

Henry Bottomley, May 23 2000

Different from highest power of 4 dividing n! (see A090616).

			  a(10^0) = 0.
  a(10^1) = 2.
  a(10^2) = 32.
  a(10^3) = 330.
  a(10^4) = 3331.
  a(10^5) = 33330.
  a(10^6) = 333330.
  a(10^7) = 3333329.
  a(10^8) = 33333328.
  a(10^9) = 333333326.

Hieronymus Fischer, Table of n, a(n) for n = 0..10000

Cf. A053737, A235127 (first differences).

Cf. A011371, A027868, A054861, A054899, A067080, A090616, A098844, A132028.

function A054893(n)
  if n eq 0 then return n;
  else return A054893(Floor(n/4)) + Floor(n/4);
  end if; return A054893;
end function;
[A054893(n): n in [0..103]]; // G. C. Greubel, Feb 09 2023

Table[t=0; p=4; While[s=Floor[n/p]; t=t+s; s>0, p *= 4]; t, {n,0,100}]
Table[Total[Floor/@(n/NestList[4#&,4,6])],{n,0,80}] (* Harvey P. Dale, Jun 12 2022 *)

a(n) = (n - sumdigits(n,4))/3; \\ Kevin Ryde, Jan 08 2024

def A054893(n):
    if (n==0): return 0
    else: return A054893(n//4) + (n//4)
[A054893(n) for n in range(104)] # G. C. Greubel, Feb 09 2023

a(n) = floor(n/4) + floor(n/16) + floor(n/64) + floor(n/256) + ...
a(n) = (n - A053737(n))/3.
From Hieronymus Fischer, Sep 15 2007: (Start)
a(n) = a(floor(n/4)) + floor(n/4).
a(4*n) = a(n) + n.
a(n*4^m) = a(n) + n*(4^m-1)/3.
a(k*4^m) = k*(4^m-1)/3, for 0 <= k < 4, m >= 0.
Asymptotic behavior:
a(n) = n/3 + O(log(n)),
a(n+1) - a(n) = O(log(n)); this follows from the inequalities below.
a(n) <= (n-1)/3; equality holds true for powers of 4.
a(n) >= (n-3)/3 - floor(log_4(n)); equality holds true for n = 4^m - 1, m>0. lim inf (n/3 - a(n)) = 1/3, for n-->oo.
lim sup (n/3 - log_4(n) - a(n)) = 0, for n-->oo.
lim sup (a(n+1) - a(n) - log_4(n)) = 0, for n-->oo.
G.f.: (1/(1-x))*Sum_{k > 0} x^(4^k)/(1-x^(4^k)). (End)
Partial sums of A235127. - R. J. Mathar, Jul 08 2021

Edited by Hieronymus Fischer, Sep 15 2007

Examples added by Hieronymus Fischer, Jun 06 2012

A239690 Base 4 sum of digits of prime(n).

Original entry on oeis.org

2, 3, 2, 4, 5, 4, 2, 4, 5, 5, 7, 4, 5, 7, 8, 5, 8, 7, 4, 5, 4, 7, 5, 5, 4, 5, 7, 8, 7, 5, 10, 5, 5, 7, 5, 7, 7, 7, 8, 8, 8, 7, 11, 4, 5, 7, 7, 10, 8, 7, 8, 11, 7, 11, 2, 5, 5, 7, 4, 5, 7, 5, 7, 8, 7, 8, 7, 4, 8, 7, 5, 8, 10, 7, 10, 11, 5, 7, 5, 7, 8, 7, 11, 7

Offset: 1

Tom Edgar, Mar 24 2014

a(n) is the rank of prime(n) in the base-4 dominance order on the natural numbers.

			The sixth prime is 13, 13 in base 4 is (3,1) so a(6)=3+1=4.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Tyler Ball and Daniel Juda, Dominance over N, Rose-Hulman Undergraduate Mathematics Journal, Vol. 13, No. 2, Fall 2013.

Cf. A007605, A014499, A053737, A239619.

Cf. A000040, A007090.

a239690 = a053737 . a000040  -- Reinhard Zumkeller, Mar 20 2015

[&+Intseq(NthPrime(n),4): n in [1..100]]; // Vincenzo Librandi, Mar 25 2014

Table[Plus @@ IntegerDigits[Prime[n], 4], {n, 1, 100}] (* Vincenzo Librandi, Mar 25 2014 *)

[sum(i.digits(base=4)) for i in primes_first_n(200)]

a(n) = A053737(A000040(n)).

A240236 Triangle read by rows: sum of digits of n in base k, for 2<=k<=n.

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 2, 3, 2, 1, 2, 2, 3, 2, 1, 3, 3, 4, 3, 2, 1, 1, 4, 2, 4, 3, 2, 1, 2, 1, 3, 5, 4, 3, 2, 1, 2, 2, 4, 2, 5, 4, 3, 2, 1, 3, 3, 5, 3, 6, 5, 4, 3, 2, 1, 2, 2, 3, 4, 2, 6, 5, 4, 3, 2, 1, 3, 3, 4, 5, 3, 7, 6, 5, 4, 3, 2, 1, 3, 4, 5, 6, 4, 2, 7, 6, 5, 4, 3, 2, 1

Offset: 2

Franklin T. Adams-Watters, Apr 02 2014

			Triangle starts:
  1
  2 1
  1 2 1
  2 3 2 1
  2 2 3 2 1
  3 3 4 3 2 1

Reinhard Zumkeller, Rows n = 2..100 of triangle, flattened

Row sums give A043306.
Cf. A000120, A053735, A053737, A007953.
See A138530 for another version.
Cf. A002260, A090623.

a240236 n k = a240236_tabl !! (n-1) !! (k-1)
a240236_row n = a240236_tabl !! (n-1)
a240236_tabl = zipWith (map . flip q)
                       [2..] (map tail $ tail a002260_tabl) where
   q b n = if n < b then n else q b n' + d where (n', d) = divMod n b
-- Reinhard Zumkeller, Apr 29 2015

Table[Total[Flatten[IntegerDigits[n,k]]],{n,20},{k,2,n}]//Flatten (* Harvey P. Dale, Jan 13 2025 *)

T(n,k) = local(r=0);if(k<2,-1,while(n>0,r+=n%k;n\=k);r)

T(n, k) = sumdigits(n, k) \\ Zhuorui He, Aug 25 2025

T(n,k) = n - (k - 1) * Sum_{i=1..floor(log_k(n))} floor(n/k^i). - Ridouane Oudra, Sep 27 2024

T(n,k) = n - (k - 1) * A090623(n,k). - Zhuorui He, Aug 25 2025

A194974 Interspersion fractally induced by A194973, a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 12, 16, 19, 20, 21, 17, 18, 22, 26, 27, 28, 23, 24, 25, 29, 34, 35, 36, 30, 31, 32, 33, 37, 43, 44, 45, 38, 40, 41, 42, 39, 46, 53, 54, 55, 47, 50, 51, 52, 48, 49, 56, 64, 65, 66, 57, 61, 62, 63, 58, 59, 60, 67, 76, 77

Offset: 1

Clark Kimberling, Sep 07 2011

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence.

			Northwest corner:
1...2...4...7...11
3...5...8...13..19
6...9...14..20..27
10..15..21..28..36
12..17..23..30..38

Cf. A194959, A194973, A194975.

p[n_] := Floor[(n + 3)/4] + Mod[n - 1, 4]
Table[p[n], {n, 1, 90}]  (* A053737(n+4), n>=0 *)
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20]  (* A194973 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13},
{k, 1, n}]]  (* A194974 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]]  (* A194975 *)

cp's OEIS Frontend

A033095 Number of 1's when n is written in base b for 2<=b<=n+1.

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Mathematica

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A053839 a(n) = (sum of digits of n written in base 4) modulo 4.

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Maple

Mathematica

PARI

PARI

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A053831 Sum of digits of n written in base 11.

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C

Mathematica

PARI

PARI

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A244041 Sum of digits of n written in fractional base 4/3.

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Sage

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A173525 a(n) = 1 + A053824(n-1), where A053824 = sum of digits in base 5.

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Haskell

Maple

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PARI

PARI

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A346688 Replace 4^k with (-1)^k in base-4 expansion of n.

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Mathematica

Python

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A054893 a(n) = Sum_{j > 0} floor(n/4^j).

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