A053824 -id:A053824 - cp's OEIS frontend

A216789 Table read by antidiagonals: T(n,k) is the digital sum of k in base n displayed in decimal.

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

Offset: 2

Henry Bottomley & Robert G. Wilson v, Sep 16 2012

T(n,k) is the least number of powers of n that add up to k. - Mohammed Yaseen, Nov 12 2022

			A000120   0, 1, 1, 2, 1, 2, 2, 3, 1, 2,  2,  3,  2,  3,  3,  4, 1, 2, 2
A053735   0, 1, 2, 1, 2, 3, 2, 3, 4, 1,  2,  3,  2,  3,  4,  3, 4, 5, 2
A053737   0, 1, 2, 3, 1, 2, 3, 4, 2, 3,  4,  5,  3,  4,  5,  6, 1, 2, 3
A053824   0, 1, 2, 3, 4, 1, 2, 3, 4, 5,  2,  3,  4,  5,  6,  3, 4, 5, 6
A053827   0, 1, 2, 3, 4, 5, 1, 2, 3, 4,  5,  6,  2,  3,  4,  5, 6, 7, 3
A053828   0, 1, 2, 3, 4, 5, 6, 1, 2, 3,  4,  5,  6,  7,  2,  3, 4, 5, 6
A053829   0, 1, 2, 3, 4, 5, 6, 7, 1, 2,  3,  4,  5,  6,  7,  8, 2, 3, 4
A053830   0, 1, 2, 3, 4, 5, 6, 7, 8, 1,  2,  3,  4,  5,  6,  7, 8, 9, 2
A007953   0, 1, 2, 3, 4, 5, 6, 7, 8, 9,  1,  2,  3,  4,  5,  6, 7, 8, 9
A053831   0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,  1,  2,  3,  4,  5, 6, 7, 8
A053832   0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,  1,  2,  3,  4, 5, 6, 7
A053833   0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,  1,  2,  3, 4, 5, 6
A053834   0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,  1,  2, 3, 4, 5
A053835   0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,  1, 2, 3, 4
A053836   0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3

Robert Israel, Table of n, a(n) for n = 2..10012 (indices corrected to start at 2 by Sidney Cadot, Jan 07 2023)
K. Atanassov, On Some of Smarandache's Problems
Eric Weisstein's World of Mathematics, Digit Sum
Wikipedia, Digit sum

Cf. A000120, A053735, A053737, A053824, A053827, A053828, A053829, A053830, A007953, A053831, A053832, A053833, A053834, A053835, A053836.

[seq(seq(convert(convert(n-b,base,b),`+`),b=n..2,-1),n=1..15)]; # Robert Israel, Aug 02 2020

DigitSum[n_, b_: 10] := Total[IntegerDigits[n, b]]; Table[ DigitSum[n - b, b], {n, 2, 13}, {b, n, 2, -1}] // Flatten

Name and offset corrected by Mohammed Yaseen, Nov 12 2022

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)).

A309956 Product of digits of (n written in base 5).

Original entry on oeis.org

0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 0, 2, 4, 6, 8, 0, 3, 6, 9, 12, 0, 4, 8, 12, 16, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 0, 2, 4, 6, 8, 0, 3, 6, 9, 12, 0, 4, 8, 12, 16, 0, 0, 0, 0, 0, 0, 2, 4, 6, 8, 0, 4, 8, 12, 16, 0, 6, 12, 18, 24, 0, 8, 16, 24, 32, 0, 0, 0, 0, 0, 0, 3, 6, 9, 12, 0, 6, 12, 18, 24, 0, 9, 18, 27, 36, 0, 12, 24, 36, 48, 0

Offset: 0

Ilya Gutkovskiy, Aug 24 2019

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A007091, A007954, A053824.

[0] cat [&*Intseq(n,5):n in [1..100]]; // Marius A. Burtea, Aug 25 2019

Table[Times @@ IntegerDigits[n, 5], {n, 0, 100}]

a(n) = my(d = if (n, digits(n,5), [0])); vecprod(d); \\ Michel Marcus, Aug 25 2019

G.f. A(x) satisfies: A(x) = x * (1 + 2*x + 3*x^2 + 4*x^3) * (1 + A(x^5)).

A194966 Interspersion fractally induced by A194965, a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 17, 22, 25, 26, 27, 28, 23, 24, 29, 33, 34, 35, 36, 30, 31, 32, 37, 42, 43, 44, 45, 38, 39, 40, 41, 46, 52, 53, 54, 55, 47, 48, 49, 50, 51, 56, 63, 64, 65, 66, 57, 59, 60, 61, 62, 58, 67, 75, 76

Offset: 1

Clark Kimberling, Sep 07 2011

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. Every pair of rows eventually intersperse. As a sequence, A194966 is a permutation of the positive integers, with inverse A194967.

			Northwest corner:
1...2...4...7...11..16
3...5...8...12..18..25
6...9...13..19..26..34
10..14..20..27..35..44
15..21..28..36..45..55

Index entries for sequences that are permutations of the natural numbers

Cf. A194959, A194965, A194967 (inverse).

p[n_] := Floor[(n + 4)/5] + Mod[n - 1, 5]
Table[p[n], {n, 1, 90}]  (* A053824(n+5), 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]   (* A194965 *)
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}]]  (* A194966 *)
q[n_] := Position[w, n]; Flatten[
Table[q[n], {n, 1, 80}]]  (* A194967 *)

A053840 (Sum of digits of n written in base 5) modulo 5.

Original entry on oeis.org

0, 1, 2, 3, 4, 1, 2, 3, 4, 0, 2, 3, 4, 0, 1, 3, 4, 0, 1, 2, 4, 0, 1, 2, 3, 1, 2, 3, 4, 0, 2, 3, 4, 0, 1, 3, 4, 0, 1, 2, 4, 0, 1, 2, 3, 0, 1, 2, 3, 4, 2, 3, 4, 0, 1, 3, 4, 0, 1, 2, 4, 0, 1, 2, 3, 0, 1, 2, 3, 4, 1, 2, 3, 4, 0, 3, 4, 0, 1, 2, 4, 0, 1, 2, 3, 0, 1, 2, 3, 4, 1, 2, 3, 4, 0, 2, 3, 4, 0, 1, 4, 0, 1, 2, 3

Offset: 0

Henry Bottomley, Mar 28 2000

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

Michael De Vlieger, 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

Cf. A010874, A053824, A141803.

Cf. A010060, A053837-A053844.

Mod[Total@ IntegerDigits[#, 5], 5] & /@ Range[0, 120] (* Michael De Vlieger, May 17 2016 *)

a(n) = vecsum(digits(n,5)) % 5; \\ Michel Marcus, May 16 2016

a(n) = A010874(A053824(n)). - Andrey Zabolotskiy, May 18 2016

A183226 Sum of digits of (2^n) in base 5, also sum of digits of (10^n) in base 5.

Original entry on oeis.org

1, 2, 4, 4, 4, 4, 8, 4, 4, 8, 12, 12, 12, 12, 8, 12, 16, 20, 20, 20, 16, 12, 20, 24, 28, 20, 32, 32, 24, 32, 40, 40, 32, 24, 28, 32, 32, 40, 28, 36, 36, 40, 44, 40, 36, 40, 36, 44, 44, 44, 44, 48, 52, 52, 48, 56, 40, 56, 68, 60, 52, 52, 48, 60, 56, 64, 60, 48, 56, 60, 60, 64, 60, 60, 60, 64, 52, 48, 64, 68, 56, 80, 80

Offset: 0

Washington Bomfim, Jan 01 2011

If i >= 2, a(n) mod 4 = 0. (Cf. A053824)

			a(9) = 8 because 10^9 = 4022000000000_5, and 2^9 = 512 = 4022_5.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A055476, A173670, A183227, A183228.

a:= n-> add(i, i=convert (2^n, base, 5)):
seq(a(n), n=0..82);  # Alois P. Heinz, Jan 06 2011

Table[Plus@@IntegerDigits[2^n, 5], {n, 0, 49}] (* Either that one or this one *) Table[Plus@@IntegerDigits[10^n, 5], {n, 0, 49}] (* Alonso del Arte, Jan 06 2011 *)

\\  L is the list of the N digits of 2^n in quinary.
     \\ L[1] = a_0 , ..., L[N] = a_(N-1).
convert(n)={n=2^n; x=n; N=floor(log(n)/log(5))+1;
L = listcreate(N);
while(x, n=floor(n/5); r=x-5*n; listput(L, r); x=n; );
L; N};
for(n=0,100,convert(n);an=0;for(i=1,N,an+=L[i];); print1(an,", "));

t(n) = if(n<1, 0, if(n%5, t(n-1)+1, t(n/5)));
vector(200, n, n--; t(2^n)) \\ Altug Alkan, Oct 28 2015

A183227 a(n) is the base-5 digit sum of 10^n-1.

Original entry on oeis.org

0, 5, 11, 15, 19, 23, 31, 31, 35, 43, 51, 55, 59, 63, 63, 71, 79, 87, 91, 95, 95, 95, 107, 115, 123, 119, 135, 139, 135, 147, 159, 163, 159, 155, 163, 171, 175, 187, 179, 191, 195, 203, 211, 211, 211, 219, 219, 231, 235, 239

Offset: 0

Washington Bomfim, Jan 01 2011

			a(9) = 43 because 10^9 - 1 is written as 4021444444444_5, and 2^9 - 1 = 511 is written as 4021_5.

Cf. A183226, A183228.

A053824 := proc(n) add(d,d=convert(n,base,5)) ; end proc:
A183227 := proc(n) A053824(10^n-1) ; end proc: # R. J. Mathar, Jan 09 2011

\\L is a list of the N digits of 2^n - 1 in  quinary
convert(n)={n = 2^n - 1; x=n; N=floor(log(n)/log(5))+1;
L = listcreate(N);
while(x, n=floor(n/5); r=x-5*n; listput(L, r); x=n; );
L; N};
print1("0, "); for(n = 1,100, convert(n); s = 0; for(i = 1, N, s += L[i];); print1(s+4*n, ", "));

a(n) = A053824(10^n-1) = 4*n + A053824(2^n-1).

A183228 a(n) is the base-5 digit sum of 10^n+1.

Original entry on oeis.org

2, 3, 5, 5, 5, 5, 9, 5, 5, 9, 13, 13, 13, 13, 9, 13, 17, 21, 21, 21, 17, 13, 21, 25, 29, 21, 33, 33, 25, 33, 41, 41, 33, 25, 29, 33, 33, 41, 29, 37, 37, 41, 45, 41, 37, 41, 37, 45, 45, 45, 45, 49, 53, 53, 49, 57, 41, 57, 69

Offset: 0

Washington Bomfim, Jan 02 2011

			a(9) = 9 because 10^9 + 1 is written as 4022000000001_5, and 2^9 = 512 is written as 4022_5.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A183226, A183227.

A053824 := proc(n) add(d,d=convert(n,base,5)) ; end proc:
A183228 := proc(n) A053824(10^n+1) ; end proc: # R. J. Mathar, Jan 09 2011

Table[Total[IntegerDigits[10^n+1,5]],{n,0,60}] (* Harvey P. Dale, Jun 10 2018 *)

\\ L is the list of the N digits of 2^n in quinary.
convert(n)={ n = 2^n; x = n; N = floor(log(n)/log(5))+1;
L = listcreate(N);
while(x, n=floor(n/5); r= x-5*n; listput(L, r); x = n; );
L; N};
for(n=0,100,convert(n); s=0;for(i=1,N, s+=L[i];); print1(s+1,", "));

a(n) = sumdigits(10^n+1, 5); \\ Michel Marcus, Sep 20 2019

a(n) = A053824(10^n+1) = 1 + A053824(2^n).

Formula corrected by Robert Israel, Sep 19 2019

A230865 a(n) = n + (sum of digits in base-5 representation of n).

Original entry on oeis.org

0, 2, 4, 6, 8, 6, 8, 10, 12, 14, 12, 14, 16, 18, 20, 18, 20, 22, 24, 26, 24, 26, 28, 30, 32, 26, 28, 30, 32, 34, 32, 34, 36, 38, 40, 38, 40, 42, 44, 46, 44, 46, 48, 50, 52, 50, 52, 54, 56, 58, 52, 54, 56, 58, 60, 58, 60, 62, 64, 66, 64, 66, 68, 70, 72, 70, 72, 74, 76, 78, 76, 78, 80, 82, 84, 78, 80, 82, 84, 86, 84

Offset: 0

N. J. A. Sloane, Nov 05 2013

The image of this sequence is the set of nonnegative even numbers (A005843). Joshi (1973) proved that the sequence of base-q self numbers (analogous to A003052) is the sequence of odd numbers (A005408) for all odd q. - Amiram Eldar, Nov 28 2020

V. S. Joshi, Ph.D. dissertation, Gujarat Univ., Ahmedabad (India), October, 1973.
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384-386.

Donovan Johnson, Table of n, a(n) for n = 0..10000

Cf. A053824, A230641.

Table[n + Plus @@ IntegerDigits[n, 5], {n, 0, 100}] (* Amiram Eldar, Nov 28 2020 *)

a(n) = n + A053824(n). - Amiram Eldar, Nov 28 2020

A096289 Sum of digits of n in bases 2 and 5.

Original entry on oeis.org

0, 2, 3, 5, 5, 3, 4, 6, 5, 7, 4, 6, 6, 8, 9, 7, 5, 7, 8, 10, 6, 8, 9, 11, 10, 4, 5, 7, 7, 9, 6, 8, 5, 7, 8, 6, 6, 8, 9, 11, 6, 8, 9, 11, 11, 9, 10, 12, 10, 12, 5, 7, 7, 9, 10, 8, 7, 9, 10, 12, 8, 10, 11, 13, 9, 7, 8, 10, 10, 12, 9, 11, 10, 12, 13, 7, 7, 9, 10, 12, 6, 8, 9, 11, 11, 9, 10, 12, 11, 13

Offset: 0

Miklos Kristof, Peter Boros, Jun 24 2004

Let n = Sum(c(k)*2^k), c(k) = 0,1, be the binary form of n, n = Sum(d(k)*5^k), d(k) = 0,1,2,3,4 the base 5 form; then a(n) = Sum(c(k)+d(k)).

a(n) mod 2 = doubled Thue-Morse sequence A095190.

			n=13: 13=1*2^3+1*2^2+1*2^0, 1+1+1=3, 13=2*5^1+3*5^0, 2+3=5, so a(13)=3+5=8.

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000120, A010059, A010060, A053824, A095190, A096288.

a[n_] := Total[Flatten@ IntegerDigits[n, {2, 5}]]; Array[a, 100, 0] (* Amiram Eldar, Jul 28 2023 *)

a(n) = A000120(n) + A053824(n). - Amiram Eldar, Jul 28 2023

cp's OEIS Frontend

A216789 Table read by antidiagonals: T(n,k) is the digital sum of k in base n displayed in decimal.

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

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A309956 Product of digits of (n written in base 5).

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A194966 Interspersion fractally induced by A194965, a rectangular array, by antidiagonals.

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A053840 (Sum of digits of n written in base 5) modulo 5.

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A183226 Sum of digits of (2^n) in base 5, also sum of digits of (10^n) in base 5.

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A183227 a(n) is the base-5 digit sum of 10^n-1.

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