A245355 -id:A245355 - cp's OEIS frontend

A024647 n written in fractional base 8/5.

0, 1, 2, 3, 4, 5, 6, 7, 50, 51, 52, 53, 54, 55, 56, 57, 520, 521, 522, 523, 524, 525, 526, 527, 570, 571, 572, 573, 574, 575, 576, 577, 5240, 5241, 5242, 5243, 5244, 5245, 5246, 5247, 5710, 5711, 5712, 5713, 5714, 5715, 5716, 5717, 5760, 5761, 5762, 5763, 5764, 5765

Offset: 0

David W. Wilson

To represent a number in base 8, if a digit exceeds 7, subtract 8 and carry 1. In the fractional base 8/5, subtract 8 and carry 5.

			The integers 0 through 7 are written with the digits 0 through 7.
Then, since b = 8/5 is written as 10, and 8 is five times 8/5, 8 is 50 in base 8/5, and therefore a(8) = 50.
a(16) = 520, since 5 * (8/5)^2 + 2 * (8/5) = 5 * 64/25 + 2 * 8/5 = 64/5 + 16/5 = 80/5 = 16.

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

Cf. A245355.

Select[Table[FromDigits[IntegerDigits[n, 8]], {n, 0, 4095}], IntegerQ[FromDigits[IntegerDigits[#], 8/5]] &] (* Alonso del Arte, Aug 05 2019 *)
a[n_] := a[n] = If[n == 0, 0, 10 * a[5 * Floor[n/8]] + Mod[n, 8]]; Array[a, 50, 0] (* Amiram Eldar, Aug 02 2025 *)

a(n) = if(n == 0, 0, 10 * a(n\8 * 5) + n % 8); \\ Amiram Eldar, Aug 02 2025

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

Original entry on oeis.org

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

A245336 Sum of digits of n written in fractional base 8/7.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 7, 8, 9, 10, 11, 12, 13, 14, 13, 14, 15, 16, 17, 18, 19, 20, 18, 19, 20, 21, 22, 23, 24, 25, 22, 23, 24, 25, 26, 27, 28, 29, 25, 26, 27, 28, 29, 30, 31, 32, 27, 28, 29, 30, 31, 32, 33, 34, 28, 29, 30, 31, 32, 33, 34, 35, 28, 29, 30, 31

Offset: 0

Hailey R. Olafson, Jul 18 2014

The base 8/7 expansion is unique and thus the sum of digits function is well-defined.

			In base 8/7 the number 14 is represented by 76 and so a(14) = 7 + 6 = 13.

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

Cf. A000120, A007953, A024649, A053829, A244040.

a[n_] := a[n] = If[n == 0, 0, a[7 * Floor[n/8]] + Mod[n, 8]]; Array[a, 100, 0] (* Amiram Eldar, Aug 02 2025 *)

a(n) = if(n == 0, 0, a(n\8 * 7) + n % 8); \\ Amiram Eldar, Aug 02 2025

# uses [basepqsum from A245355]
[basepqsum(8,7,w) for w in [0..200]]

a(n) = A007953(A024649(n)).

A245337 Sum of digits of n in fractional base 7/6.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 12, 11, 12, 13, 14, 15, 16, 17, 15, 16, 17, 18, 19, 20, 21, 18, 19, 20, 21, 22, 23, 24, 20, 21, 22, 23, 24, 25, 26, 21, 22, 23, 24, 25, 26, 27, 21, 22, 23, 24, 25, 26, 27, 27, 28, 29, 30, 31, 32, 33, 26, 27, 28, 29, 30

Offset: 0

James Van Alstine, Jul 18 2014

The base 7/6 expansion is unique and thus the sum of digits function is well-defined.

			In base 7/6 the number 7 is represented by 60 and so a(7) = 6+0 = 6.

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

Cf. A024643, A007953, A053828.

a[n_] := a[n] = If[n == 0, 0, a[6 * Floor[n/7]] + Mod[n, 7]]; Array[a, 100, 0] (* Amiram Eldar, Jul 31 2025 *)

a(n) = if(n == 0, 0, a(n\7 * 6) + n % 7); \\ Amiram Eldar, Jul 31 2025

# uses [basepqsum from A245355]
[basepqsum(7,6,y) for y in [0..200]]

a(n) = A007953(A024643(n)). - Amiram Eldar, Jul 31 2025

A245338 Sum of digits of n written in fractional base 9/8.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 8, 9, 10, 11, 12, 13, 14, 15, 16, 15, 16, 17, 18, 19, 20, 21, 22, 23, 21, 22, 23, 24, 25, 26, 27, 28, 29, 26, 27, 28, 29, 30, 31, 32, 33, 34, 30, 31, 32, 33, 34, 35, 36, 37, 38, 33, 34, 35, 36, 37, 38, 39, 40, 41, 35, 36, 37, 38, 39

Offset: 0

Tom Edgar, Jul 18 2014

The base 9/8 expansion is unique and thus the sum of digits function is well-defined.

			In base 9/8 the number 16 is represented by 87 and so a(16) = 8 + 7 = 15.

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

Cf. A000120, A007953, A024656, A053830, A244040.

a[n_] := a[n] = If[n == 0, 0, a[8 * Floor[n/9]] + Mod[n, 9]]; Array[a, 100, 0] (* Amiram Eldar, Aug 04 2025 *)

a(n) = if(n == 0, 0, a(n\9 * 8) + n % 9); \\ Amiram Eldar, Aug 04 2025

# uses [basepqsum from A245355]
[basepqsum(9,8,i) for i in [0..100]]

a(n) = A007953(A024656(n)).

A245339 Sum of digits of n written in fractional base 10/9.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 39, 40, 41, 42, 43, 44, 45, 46

Offset: 0

Hailey R. Olafson, Jul 18 2014

The base 10/9 expansion is unique and thus the sum of digits function is well-defined.

			In base 10/9 the number 14 is represented by 94 and so a(14) = 9 + 4 = 13.

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

Cf. A000120, A007953, A024664, A244040.

a[n_] := a[n] = If[n == 0, 0, a[9 * Floor[n/10]] + Mod[n, 10]]; Array[a, 100, 0] (* Amiram Eldar, Aug 04 2025 *)

a(n) = if(n == 0, 0, a(n\10 * 9) + n % 10); \\ Amiram Eldar, Aug 04 2025

# uses [basepqsum from A245355]
[basepqsum(10,9,w) for w in [0..200]]

a(n) = A007953(A024664(n)).

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

A245342 Sum of digits of n written in fractional base 7/2.

Original entry on oeis.org

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

Offset: 0

Hailey R. Olafson, Jul 18 2014

The base 7/2 expansion is unique and thus the sum of digits function is well-defined.

			In base 7/2 the number 14 is represented by 40 and so a(14) = 4 + 0 = 4.

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

Cf. A007953, A000120, A024639, A053828, A244040.

a[n_] := a[n] = If[n == 0, 0, a[2 * Floor[n/7]] + Mod[n, 7]]; Array[a, 100, 0] (* Amiram Eldar, Jul 31 2025 *)

a(n) = if(n == 0, 0, a(n\7 * 2) + n % 7); \\ Amiram Eldar, Jul 31 2025

# uses [basepqsum from A245355]
[basepqsum(7,2,w) for w in [0..200]]

a(n) = A007953(A024639(n)).

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

A245344 Sum of digits of n written in fractional base 7/3.

Original entry on oeis.org

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

Offset: 0

James Van Alstine, Jul 18 2014

The base 7/3 expansion is unique, and thus the sum of digits function is well-defined.

			In base 7/3 the number 7 is represented by 30 and so a(7) = 3 + 0 = 3.

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

Cf. A007953, A024640, A053828, A007953.

a:= proc(n) `if`(n<1, 0, irem(n, 7, 'q')+a(3*q)) end:
seq(a(n), n=0..69);  # Alois P. Heinz, May 14 2021

a[n_] := a[n] = If[n == 0, 0, a[3 * Floor[n/7]] + Mod[n, 7]]; Array[a, 100, 0] (* Amiram Eldar, Aug 02 2025 *)

a(n) = if(n == 0, 0, a(n\7 * 3) + n % 7); \\ Amiram Eldar, Aug 02 2025

# uses [basepqsum from A245355]
[basepqsum(7,3,y) for y in [0..200]]

a(n) = A007953(A024640(n)). - Amiram Eldar, Aug 02 2025

Definition corrected by Georg Fischer, May 14 2021

cp's OEIS Frontend

A024647 n written in fractional base 8/5.

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

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A245336 Sum of digits of n written in fractional base 8/7.

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A245337 Sum of digits of n in fractional base 7/6.

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A245338 Sum of digits of n written in fractional base 9/8.

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A245339 Sum of digits of n written in fractional base 10/9.

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

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