A053824 -id:A053824 - cp's OEIS frontend

A245335 Sum of digits of n in fractional base 5/4.

0, 1, 2, 3, 4, 4, 5, 6, 7, 8, 7, 8, 9, 10, 11, 9, 10, 11, 12, 13, 10, 11, 12, 13, 14, 10, 11, 12, 13, 14, 14, 15, 16, 17, 18, 13, 14, 15, 16, 17, 16, 17, 18, 19, 20, 14, 15, 16, 17, 18, 16, 17, 18, 19, 20, 20, 21, 22, 23, 24, 17, 18, 19, 20, 21, 18, 19, 20, 21

Offset: 0

James Van Alstine, Jul 18 2014

The base 5/4 expansion is unique and thus the sum of digits function is well-defined.

			In base 5/4 the number 7 is represented by 42 and so a(7) = 4+2 = 6.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for sequences related to fractional bases.

Cf. A024634, A007953, A053824.

a[n_] := a[n] = If[n == 0, 0, a[4 * Floor[n/5]] + Mod[n, 5]]; Array[a, 100, 0] (* Amiram Eldar, Jul 31 2025 *)

a(n) = my(ret=0,r); while(n, [n,r]=divrem(n,5); ret+=r; n<<=2); ret; \\ Kevin Ryde, Aug 11 2023

# uses [basepqsum from A245355]
[basepqsum(5,4,y) for y in [0..200]]

a(n) = A007953(A024634(n)). - Kevin Ryde, Aug 11 2023

A245341 Sum of digits of n written in fractional base 5/2.

Original entry on oeis.org

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

Offset: 0

James Van Alstine, Jul 18 2014

The base 5/2 expansion is unique and thus the sum of digits function is well-defined.

			In base 5/2 the number 7 is represented by 22 and so a(7) = 2+2 = 4.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Index entries for sequences related to fractional bases.

Cf. A024632, A007953, A053824, A215088.

a:= proc(n) `if`(n<1, 0, irem(n, 5, 'q')+a(2*q)) end:
seq(a(n), n=0..81);  # Alois P. Heinz, May 14 2021

a[n_] := a[n] = If[n == 0, 0, a[2 * Floor[n/5]] + Mod[n, 5]]; Array[a, 100, 0] (* Amiram Eldar, Jul 30 2025 *)

a(n) = if(n == 0, 0, a(n\5 * 2) + n % 5); \\ Amiram Eldar, Jul 30 2025

# uses [basepqsum from A245355]
[basepqsum(5,2,y) for y in [0..200]]

a(n) = A007953(A024632(n)). - Amiram Eldar, Jul 30 2025

Definition corrected by Georg Fischer, May 14 2021

A245343 Sum of digits of n written in fractional base 5/3.

Original entry on oeis.org

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

Offset: 0

James Van Alstine, Jul 18 2014

The base 5/3 expansion is unique and thus the sum of digits function is well-defined.

			In base 5/3 the number 7 is represented by 32 and so a(7) = 3+2 = 5.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for sequences related to fractional bases.

Cf. A024633, A007953, A053824.

a[n_] := a[n] = If[n == 0, 0, a[3 * Floor[n/5]] + Mod[n, 5]]; Array[a, 100, 0] (* Amiram Eldar, Jul 30 2025 *)

a(n) = if(n == 0, 0, a(n\5 * 3) + n % 5); \\ Amiram Eldar, Jul 30 2025

# uses [basepqsum from A245355]
[basepqsum(5,3,y) for y in [0..200]]

a(n) = A007953(A024633(n)). - Amiram Eldar, Jul 30 2025

Name corrected by Bernard Schott, Mar 18 2020

A000999 5-adic valuation of binomial(2n,n): largest k such that 5^k divides binomial(2n, n).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, R. K. Guy

T. D. Noe, Table of n, a(n) for n = 0..1000
E. E. Kummer, Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen, Journal für die reine und angewandte Mathematik, Vol. 44 (1852), pp. 93-146; alternative link.
Dorel Miheţ, Legendre's and Kummer's theorems again, Resonance, Vol. 15, No. 12 (2010), pp. 1111-1121; alternative link.
Armin Straub, Victor H. Moll and Tewodros Amdeberhan, The p-adic valuation of k-central binomial coefficients, Acta Arithmetica, Vol. 140, No. 1 (2009), pp. 31-42.
Wikipedia, Kummer's theorem.

Cf. A000984, A000989, A053824, A112765.

Table[IntegerExponent[Binomial[2*n, n], 5], {n, 0, 100}] (* T. D. Noe, Jun 21 2012 *)

a(n)=if(n<0,0,valuation(binomial(2*n,n),5))

a(n) = my(v=digits(n,5),c=0); sum(i=0,#v-1, c=(c+v[#v-i]>=3)); \\ Kevin Ryde, Mar 07 2023

From Amiram Eldar, Feb 12 2021: (Start)
a(n) = A112765(A000984(n)).
a(n) = (2*A053824(n) - A053824(2*n))/4. (End)

More terms from Michael Somos, Jun 27 2002

A037310 Numbers whose base-3 and base-5 expansions have the same digit sum.

Original entry on oeis.org

1, 2, 6, 7, 8, 10, 11, 15, 16, 17, 20, 30, 31, 32, 40, 41, 55, 56, 60, 61, 62, 65, 70, 71, 78, 79, 100, 101, 105, 106, 107, 129, 135, 136, 137, 141, 142, 143, 145, 146, 153, 154, 159, 170, 177, 178, 179, 180, 181, 182, 186, 187, 188

Offset: 1

Clark Kimberling

Also, numbers n such that the exponent of the largest power of 3 dividing n! is exactly twice the exponent of the largest power of 5 dividing n!. - Ivan Neretin, May 21 2015

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A037301 (similar, based upon 2 and 3).

Cf. A053735, A053824.

select(t -> convert(convert(t,base,3),`+`)=convert(convert(t,base,5),`+`), [$1..1000]); # Robert Israel, May 21 2015

Select[Range[200],Total[IntegerDigits[#,3]]==Total[IntegerDigits[#,5]]&] (* Harvey P. Dale, Jun 06 2016 *)

is(n)=sumdigits(n,3)==sumdigits(n,5) \\ Charles R Greathouse IV, May 21 2015

A053735(a(n)) = A053824(a(n)). - Robert Israel, May 21 2015

A037316 Numbers whose base-4 and base-5 expansions have the same digit sum.

Original entry on oeis.org

1, 2, 3, 28, 29, 40, 41, 42, 43, 52, 53, 54, 76, 77, 78, 79, 90, 91, 100, 101, 102, 103, 115, 136, 137, 138, 139, 160, 161, 162, 163, 188, 189, 210, 211, 236, 237, 238, 239, 270, 271, 280, 281, 282, 283, 295, 305, 306, 307, 330, 331

Offset: 1

Clark Kimberling

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A053737 (base-4 sum of digits), A053824 (base-5 sum of digits).

filter:= n -> convert(convert(n,base,4),`+`)=convert(convert(n,base,5),`+`):
select(filter, [$1..1000]); # Robert Israel, Mar 11 2018

Select[Range[400],Total[IntegerDigits[#,4]]==Total[IntegerDigits[#,5]]&] (* Harvey P. Dale, Sep 15 2018 *)

isok(k) = sumdigits(k, 4) == sumdigits(k, 5); \\ Michel Marcus, Jun 02 2021

A037322 Numbers whose base-5 and base-6 expansions have the same digit sum.

Original entry on oeis.org

1, 2, 3, 4, 45, 46, 47, 66, 67, 68, 69, 85, 86, 87, 88, 89, 145, 146, 147, 148, 149, 168, 169, 186, 187, 188, 189, 225, 226, 227, 265, 266, 267, 268, 269, 306, 307, 308, 309, 325, 326, 327, 328, 329, 370, 371, 408, 409, 490, 491

Offset: 1

Clark Kimberling

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Select[Range[500],Total[IntegerDigits[#,5]]==Total[IntegerDigits[#,6]]&] (* Harvey P. Dale, Mar 27 2022 *)

{n: A053824(n) = A053827(n).} - R. J. Mathar, Jun 30 2021

A135738 Least positive integer with even digit sum in bases 2..n.

Original entry on oeis.org

3, 6, 10, 10, 54, 54, 54, 54, 130, 130, 130, 130, 390, 390, 2000, 2000, 3238, 3238, 4080, 4080, 7326, 7326, 16584, 16584, 17310, 17310, 17310, 17310, 17310, 17310, 17310, 17310, 231000, 231000, 231000, 231000, 466352, 466352, 466352, 466352, 3020830

Offset: 2

M. F. Hasler, Dec 06 2007

The sequence is obviously increasing. It seems that a(2n+1) = a(2n) for n > 1. Is there a simple proof? Is there a simple way to construct a(n)? Notice the pattern in base N, e.g., 130 = 10000010_2 = 11211_3 = 2002_4 = 1010_5 = 334_6 = 244_7 = 202_8 = 154_9 = 109_11 = {10}{10}_12 = {10}0_13.

			a(2)=3 since 1=1_2, 2=10_2, so 3=11_2 is the number > 0 with even digit sum (1+1) in base 2.
a(3)=6 since 4=100_2, 5=12_3, so 6=20_3=110_2 is the least N > 0 with even digit sum in base 2 and in base 3.
a(4)=a(5)=10=1010_2=101_3=22_4=20_5 is the least N > 0 having even digit sum in bases 2 through 4 and has so also in base 5.

"Davar55" on mersenneforum.org, Puzzles / "Sum of digits".

Cf. A000120, A053735, A053737, A053824, A053827-A053836, A007953.

digitsum(n,b=10,s)={n=[n];while(n=divrem(n[1],b),s+=n[2]);s}
A135738(Bmax,n=1)={until(!n++,for(b=2,Bmax,digitsum(n,b)%2&next(2));return(n))} /* n-th element of the sequence */
t=1;for(b=2,100,print(b,":",t=A135738(b,t))) /* display the list */

Corrected example a(3)=5 to a(3)=6 David Yablon (davar55(AT)yahoo.com), Mar 19 2010

cp's OEIS Frontend

A245335 Sum of digits of n in fractional base 5/4.

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A245341 Sum of digits of n written in fractional base 5/2.

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A245343 Sum of digits of n written in fractional base 5/3.

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A000999 5-adic valuation of binomial(2*n,n): largest k such that 5^k divides binomial(2*n, n).

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A037310 Numbers whose base-3 and base-5 expansions have the same digit sum.

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Maple

Mathematica

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A037316 Numbers whose base-4 and base-5 expansions have the same digit sum.

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Maple

Mathematica

PARI

A037322 Numbers whose base-5 and base-6 expansions have the same digit sum.

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A000999 5-adic valuation of binomial(2n,n): largest k such that 5^k divides binomial(2n, n).