A239690 -id:A239690 - cp's OEIS frontend

A053737 Sum of digits of (n written in base 4).

0, 1, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 5, 6, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 5, 6, 4, 5, 6, 7, 2, 3, 4, 5, 3, 4, 5, 6, 4, 5, 6, 7, 5, 6, 7, 8, 3, 4, 5, 6, 4, 5, 6, 7, 5, 6, 7, 8, 6, 7, 8, 9, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 5, 6, 4, 5, 6, 7, 2, 3, 4, 5, 3, 4, 5, 6, 4, 5, 6, 7, 5, 6, 7, 8, 3, 4, 5, 6, 4, 5, 6, 7, 5

Offset: 0

Henry Bottomley, Mar 28 2000

Also the fixed point of the morphism 0->{0,1,2,3}, 1->{1,2,3,4}, 2->{2,3,4,5}, etc. - Robert G. Wilson v, Jul 27 2006

			a(20) = 1+1+0 = 2 because 20 is written as 110 base 4.
From _Omar E. Pol_, Feb 21 2010: (Start)
This can be written as a triangle (cf. A000120):
  0,
  1,2,3,
  1,2,3,4,2,3,4,5,3,4,5,6,
  1,2,3,4,2,3,4,5,3,4,5,6,4,5,6,7,2,3,4,5,3,4,5,6,4,5,6,7,5,6,7,8,3,4,5,6,4,5,6,7,5,6,7,8,6,7,8,9,
  1,2,3,4,2,3,4,5,3,4,5,6,4,5,6,7,2,3,4,5,3,4,5,6,4,5,6,7,5,6,7,8,3,4,5,6,4,...
where the rows converge to A173524.
(End)

Donovan Johnson, Table of n, a(n) for n = 0..10000
F. T. Adams-Watters and F. Ruskey, Generating Functions for the Digital Sum and Other Digit Counting Sequences, JIS Vol. 12 (2009), Article 09.5.6.
Steve Butler and Ron L. Graham, Shuffling with ordered cards, arXiv 1003:4422 [math.CO], 2010.
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. A231664-A231667, A138530.
Cf. A173524. - Omar E. Pol, Feb 21 2010
Sum of digits of n written in bases 2-16: A000120, A053735, this sequence, A053824, A053827, A053828, A053829, A053830, A007953, A053831, A053832, A053833, A053834, A053835, A053836.
Related base-4 sequences: A053737, A230631, A230632, A010064, A230633, A230634, A230635, A230636, A230637, A230638, A010065 (trajectory of 1).
Cf. A007090, A239690.

a053737 n = if n == 0 then 0 else a053737 m + r where (m, r) = divMod n 4
-- Reinhard Zumkeller, Mar 19 2015

for u=0:104; sol(u+1)=sum(dec2base(u,4)-'0');end
sol % Marius A. Burtea, Jan 17 2019

[&+Intseq(n,4):n in [0..104]]; // Marius A. Burtea, Jan 17 2019

A053737 := proc(n)
    add(d,d=convert(n,base,4)) ;
end proc: # R. J. Mathar, Oct 31 2012

Table[Plus @@ IntegerDigits[n, 4], {n, 0, 100}] (* or *)
Nest[ Flatten[ #1 /. a_Integer -> {a, a+1, a+2, a+3}] &, {0}, 4] (* Robert G. Wilson v, Jul 27 2006 *)
DigitSum[Range[0, 100], 4] (* Paolo Xausa, Aug 01 2024 *)

PARI
```
a(n)=if(n<1,0,if(n%4,a(n-1)+1,a(n/4)))
```

a(n) = sumdigits(n, 4); \\ Michel Marcus, Aug 24 2019

From Benoit Cloitre, Dec 19 2002: (Start)
a(0) = 0, a(4n+i) = a(n)+i for 0 <= i <= 3.
a(n) = n - 3*Sum_{k>0} floor(n/4^k) = n - 3*A054893(n). (End)
G.f.: (Sum_{k>=0} (x^(4^k) + 2*x^(2*4^k) + 3*x^(3*4^k))/(1 + x^(4^k) + x^(2*4^k) + x^(3*4^k)))/(1-x). - Franklin T. Adams-Watters, Nov 03 2005
a(n) = A138530(n,4) for n > 3. - Reinhard Zumkeller, Mar 26 2008
a(n) = Sum_{k>=0} A030386(n,k). - Philippe Deléham, Oct 21 2011
a(n) = A007953(A007090(n)). - Reinhard Zumkeller, Mar 19 2015
a(0) = 0; a(n) = a(n - 4^floor(log_4(n))) + 1. - Ilya Gutkovskiy, Aug 23 2019
Sum_{n>=1} a(n)/(n*(n+1)) = 4*log(4)/3 (Shallit, 1984). - Amiram Eldar, Jun 03 2021

A239691 Base 5 sum of digits of prime(n).

Original entry on oeis.org

2, 3, 1, 3, 3, 5, 5, 7, 7, 5, 3, 5, 5, 7, 7, 5, 7, 5, 7, 7, 9, 7, 7, 9, 9, 5, 7, 7, 9, 9, 3, 3, 5, 7, 9, 3, 5, 7, 7, 9, 7, 5, 7, 9, 9, 11, 7, 11, 7, 9, 9, 11, 9, 3, 5, 7, 9, 7, 5, 5, 7, 9, 7, 7, 9, 9, 7, 9, 11, 13, 9, 11, 11, 13, 7, 7, 9, 9, 5, 9, 11, 9, 7, 9

Offset: 1

Tom Edgar, Mar 24 2014

a(n) is the rank of prime(n) in the base-5 dominance order on the natural numbers.

			The fifth prime is 11, 11 in base 5 is (2,1) so a(5)=2+1=3.

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, A053824, A239690.

[&+Intseq(NthPrime(n),5): n in [1..100]]; // Vincenzo Librandi, Mar 25 2014

Table[Plus @@ IntegerDigits[Prime[n], 5], {n, 1, 100}] (* Vincenzo Librandi, Mar 25 2014 *)

a(n) = sumdigits(prime(n), 5); \\ Michel Marcus, Mar 04 2023

[sum(i.digits(base=5)) for i in primes_first_n(200)]

a(n) = A053824(A000040(n)).

A239692 Base 6 sum of digits of prime(n).

Original entry on oeis.org

2, 3, 5, 2, 6, 3, 7, 4, 8, 9, 6, 2, 6, 3, 7, 8, 9, 6, 7, 11, 3, 4, 8, 9, 7, 11, 8, 12, 4, 8, 7, 11, 12, 9, 9, 6, 7, 8, 12, 13, 14, 6, 11, 8, 12, 9, 11, 3, 7, 4, 8, 9, 6, 11, 7, 8, 9, 6, 7, 11, 8, 8, 7, 11, 8, 12, 6, 7, 12, 9, 13, 14, 7, 8, 9, 13, 14, 7, 11, 9

Offset: 1

Tom Edgar, Mar 24 2014

a(n) is the rank of prime(n) in the base-6 dominance order on the natural numbers.

			The sixth prime is 13, 13 in base 6 is (2,1) so a(6)=2+1=3.

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, A053827, A239690, A239691.

[&+Intseq(NthPrime(n),6): n in [1..100]]; // Vincenzo Librandi, Mar 25 2014

Table[Plus @@ IntegerDigits[Prime[n], 6], {n, 1, 100}] (* Vincenzo Librandi, Mar 25 2014 *)

a(n) = sumdigits(prime(n), 6); \\ Michel Marcus, Mar 04 2023

[sum(i.digits(base=6)) for i in primes_first_n(200)]

a(n) = A053827(A000040(n)).

A239693 Base 7 sum of digits of prime(n).

Original entry on oeis.org

2, 3, 5, 1, 5, 7, 5, 7, 5, 5, 7, 7, 11, 7, 11, 5, 5, 7, 7, 5, 7, 7, 11, 11, 13, 5, 7, 5, 7, 5, 7, 11, 11, 13, 5, 7, 7, 7, 11, 11, 11, 13, 11, 13, 5, 7, 7, 13, 11, 13, 11, 11, 13, 11, 11, 11, 11, 13, 13, 11, 13, 17, 13, 11, 13, 11, 13, 13, 5, 7, 5, 5, 7, 7

Offset: 1

Tom Edgar, Mar 24 2014

a(n) is the rank of prime(n) in the base-7 dominance order on the natural numbers.

			The fifth prime is 11, 11 in base 7 is (1,4) so a(5)=1+4=5.

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, A053828, A239690, A239691, A239692.

[&+Intseq(NthPrime(n),7): n in [1..100]]; // Vincenzo Librandi, Mar 25 2014

Table[Plus @@ IntegerDigits[Prime[n], 7], {n, 1, 100}] (* Vincenzo Librandi, Mar 25 2014 *)

a(n) = sumdigits(prime(n), 7); \\ Michel Marcus, Mar 04 2023

[sum(i.digits(base=7)) for i in primes_first_n(200)]

a(n) = A053828(A000040(n)).

A239694 Base 8 sum of digits of prime(n).

Original entry on oeis.org

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

Offset: 1

Tom Edgar, Mar 24 2014

a(n) is the rank of prime(n) in the base-8 dominance order on the natural numbers.

			The sixth prime is 13, 13 in base 8 is (1,5) so a(6)=1+5=6.

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, A053829, A239690, A239691, A239692, A239693.

[&+Intseq(NthPrime(n),8): n in [1..100]]; // Vincenzo Librandi, Mar 25 2014

Table[Plus @@ IntegerDigits[Prime[n], 8], {n, 1, 100}] (* Vincenzo Librandi, Mar 25 2014 *)

a(n) = sumdigits(prime(n), 8); \\ Michel Marcus, Mar 04 2023

[sum(i.digits(base=8)) for i in primes_first_n(200)]

a(n) = A053829(A000040(n)).

A239695 Base 9 sum of digits of prime(n).

Original entry on oeis.org

2, 3, 5, 7, 3, 5, 9, 3, 7, 5, 7, 5, 9, 11, 7, 13, 11, 13, 11, 15, 9, 15, 3, 9, 9, 5, 7, 11, 5, 9, 7, 11, 9, 11, 13, 15, 13, 3, 7, 5, 11, 5, 7, 9, 13, 7, 11, 15, 11, 13, 17, 15, 17, 11, 9, 7, 13, 7, 13, 9, 11, 13, 11, 15, 17, 13, 11, 9, 11, 13, 9, 15, 15, 13, 11

Offset: 1

Tom Edgar, Mar 24 2014

a(n) is the rank of prime(n) in the base-9 dominance order on the natural numbers.

			The fifth prime is 11, 11 in base 9 is (1,2) so a(5)=1+2=3.

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, A053830, A239690, A239691, A239692, A239693, A239694.

[&+Intseq(NthPrime(n),9): n in [1..100]]; // Vincenzo Librandi, Mar 25 2014

Table[Plus @@ IntegerDigits[Prime[n], 9], {n, 1, 100}] (* Vincenzo Librandi, Mar 25 2014 *)

a(n) = sumdigits(prime(n), 9); \\ Michel Marcus, Mar 04 2023

[sum(i.digits(base=9)) for i in primes_first_n(200)]

a(n) = A053830(A000040(n)).

cp's OEIS Frontend

A053737 Sum of digits of (n written in base 4).

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A239691 Base 5 sum of digits of prime(n).

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A239692 Base 6 sum of digits of prime(n).

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A239693 Base 7 sum of digits of prime(n).

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A239694 Base 8 sum of digits of prime(n).

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A239695 Base 9 sum of digits of prime(n).

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