A007091 -id:A007091 - cp's OEIS frontend

A024657 n written in fractional base 10/2.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 220, 221, 222, 223, 224, 225, 226, 227

Offset: 0

David W. Wilson

To represent a number in base b, if a digit exceeds b-1, subtract b and carry 1. In fractional base a/b, subtract a and carry b.

Also numbers which are written the same in base 20/2 as in base 10. The sequence consists of numbers which have digits in {0,2,4,6,8} except that the unit digit can be any from {0,1,2,3,4,5,6,7,8,9} - Henry Bottomley, Nov 17 2000

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

Cf. A007091, A058185, A058186, A102491, A118761.

a[n_] := a[n] = If[n == 0, 0, 10 * a[2 * Floor[n/10]] + Mod[n, 10]]; Array[a, 50, 0] (* Amiram Eldar, Aug 02 2025 *)

a(n) = if(n == 0, 0, 10 * a(n\10 * 2) + n % 10); \\ Amiram Eldar, Aug 02 2025

a(n) = A118761(n+1) for n < 50. - Reinhard Zumkeller, May 01 2006

A037462 a(n) = Sum_{i = 0..m} d(i)8^i, where Sum_{i = 0..m} d(i)4^i is the base 4 representation of n.

Original entry on oeis.org

0, 1, 2, 3, 8, 9, 10, 11, 16, 17, 18, 19, 24, 25, 26, 27, 64, 65, 66, 67, 72, 73, 74, 75, 80, 81, 82, 83, 88, 89, 90, 91, 128, 129, 130, 131, 136, 137, 138, 139, 144, 145, 146, 147, 152, 153, 154, 155, 192, 193, 194, 195, 200, 201, 202, 203, 208, 209, 210

Offset: 0

Clark Kimberling

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A037454, A007091, A102491.

seq(n + (1/2)*add(8^k*floor(n/4^k), k = 1..floor(ln(n)/ln(4))), n = 1..100); # Peter Bala, Dec 01 2016

Table[FromDigits[RealDigits[n, 4], 8], {n, 0, 100}]
(* Clark Kimberling, Aug 14 2012 *)

From Peter Bala, Dec 01 2016: (Start):
a(n) = n + 1/2*Sum_{k >= 1} 8^k*floor(n/4^k). Cf. A037454, A007091 and A102491.
a(0) = 0; a(n) = 8*a(n/4) if n == 0 (mod 4) else a(n) = a(n-1) + 1. (End)

Offset changed to 0 by Clark Kimberling, Aug 14 2012

A083899 Number of divisors of n with largest digit <= 4 (base 10).

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, May 08 2003

Cf. A054055, A000005, A007091, A083891, A083892, A083898, A083900.

a[n_] := DivisorSum[n, 1 &, Max[IntegerDigits[#]] <= 4 &]; Array[a, 100] (* Amiram Eldar, Jan 04 2024 *)

a(n) = A083898(n) + A083891(n) = A083900(n) - A083892(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} 1/A007091(k) = 3.92521682598923397031... . - Amiram Eldar, Jan 04 2024

A321882 a(n) is the least base b > 1 such that the sum n + n can be computed without carry.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Nov 20 2018

Equivalently, a(n) is the least base b > 1 where:
- twice the greatest digit of n is < b,
- twice the digital sum of n equals the digital sum of twice n.
The sequence is well defined as, for any n > 0, n + n can be computed without carry in base 2*n + 1.
The sequence is unbounded; by contradiction:
- suppose that v = a(n) is the greatest term of the sequence,
- we can assume that v > 2,
- let d be the greatest digit of v!^A000120(n) in base v,
- let k = floor((v-1) / d),
- necessarily a(n + k * (v!^A000120(n))) > v, QED.

			For n = 42:
- in base 2, 42 + 42 cannot be computed without carry: "101010" + "101010" = "1010100",
- in base 3, 42 + 42 cannot be computed without carry: "1120" + "1120" = "10010",
- in base 4, 42 + 42 cannot be computed without carry: "222" + "222" = "1110",
- in base 5, 42 + 42 cannot be computed without carry: "132" + "132" = "314",
- in base 6, 42 + 42 can be computed without carry: "110" + "110" = "220",
- hence a(42) = 6.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Colored scatterplot of (n, a(n)) for n = 0..10000000 (where the color is function of the initial digit of n in base a(n))
Index entries for sequences related to carryless arithmetic

See A319478 for the multiplicative variant.

Cf. A005836, A007091.

Array[Block[{b = 2}, While[2 Max@ IntegerDigits[#, b] >= b, b++]; b] &, 84, 0] (* Michael De Vlieger, Nov 25 2018 *)

a(n) = for (b=2, oo, if (2*sumdigits(n, b)==sumdigits(n*2, b), return (b)))

a(n) = 2 iff n = 0.
a(n) = 3 iff n > 0 and n belongs to A005836.
a(n * a(n)) <= a(n).
a(A007091(n)) <= 10 for any n >= 0.

A004688 Fibonacci numbers written in base 5.

Original entry on oeis.org

0, 1, 1, 2, 3, 10, 13, 23, 41, 114, 210, 324, 1034, 1413, 3002, 4420, 12422, 22342, 40314, 113211, 204030, 322241, 1031321, 1404112, 2440433, 4400100, 12341033, 22241133, 40132221, 112423404, 203111130

Offset: 0

N. J. A. Sloane

Sequence of last digit has period length of A001175(5)=20. Digits are almost evenly distributed. - Carmine Suriano, Mar 30 2012

Carmine Suriano and Vincenzo Librandi, Table of n, a(n) for n = 0..1000 (Carmine Suriano for n = 0..73)

Cf. A000045 (Fibonacci), A007091 (numbers in base 5).

[Seqint(Intseq(Fibonacci(n),5)): n in [0..50]]; // G. C. Greubel, Oct 09 2018

read("transforms") :
A004688 := proc(n)
        convert( combinat[fibonacci](n),base,5) ;
        ListTools[Reverse](%) ;
        digcatL(%) ;
end proc: # R. J. Mathar, Apr 01 2012

Table[BaseForm[Fibonacci[n],5],{n,1,20,1}] (* Vladimir Joseph Stephan Orlovsky, Jul 23 2008 *)
FromDigits[IntegerDigits[#, 5]]& / @Fibonacci[Range[0, 50]] (* Vincenzo Librandi, Jun 07 2013 *)

vector(50, n, n--; fromdigits(digits(fibonacci(n), 5))) \\ G. C. Greubel, Oct 09 2018

A032543 Numbers that, when expressed in base 5 and then interpreted in base 10, yield a multiple of the original number.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 10, 15, 20, 25, 50, 75, 100, 125, 160, 224, 237, 250, 320, 330, 375, 490, 500, 625, 800, 1000, 1120, 1185, 1250, 1600, 1650, 1875, 2450, 2500, 3125, 3800, 4000, 4704, 5000, 5600, 5925, 6250, 7600, 8000, 8250, 9375, 10000, 12250

Offset: 1

Patrick De Geest, Apr 15 1998

From Robert Israel, Apr 10 2016: (Start)
n for which n divides A007091(n).
If n is in the sequence, then so is 5*n. (End)

			25 in base 5 is 100, which interpreted in base 10 is 100 = 4 * 25.
224 in base 5 is 1344, which interpreted in base 10 is 1344 = 6 * 224.

Giovanni Resta, Table of n, a(n) for n = 1..577 (terms < 5*10^12, first 147 terms from Robert Israel)

Cf. A007091, A032542, A032544.

filter:= proc(n) local L,i;
L:= convert(n,base,5);
add(L[i]*10^(i-1),i=1..nops(L)) mod n = 0
end proc:
0, op(select(filter, [$1..10^5])); # Robert Israel, Apr 10 2016

Select[Range[0,13000], Divisible[FromDigits[IntegerDigits[#, 5]], #] &] (* Harvey P. Dale, Feb 01 2011 *)

Example and better description from Erich Friedman, Jul 21 2001
Edited by Erich Friedman, Feb 09 2002
Offset changed and 0 inserted by Robert Israel, Apr 11 2016
Name edited by Jon E. Schoenfield, Oct 25 2019

A032860 Numbers whose base-5 representation Sum_{i=0..m} d(i)*5^i has d(m) > d(m-1) < d(m-2) > ...

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 10, 11, 15, 16, 17, 20, 21, 22, 23, 26, 27, 28, 29, 51, 52, 53, 54, 57, 58, 59, 76, 77, 78, 79, 82, 83, 84, 88, 89, 101, 102, 103, 104, 107, 108, 109, 113, 114, 119, 130, 135, 136, 140, 141, 142, 145, 146, 147, 148, 255

Offset: 1

Clark Kimberling

Cf. A007091.
Cf. A032858..A032865 for bases 3..10.
Cf. A306106..A306111 and A297147 for bases 3..9 and 10.

a(1)=0 inserted by Georg Fischer, Dec 18 2020

A032912 Numbers whose set of base-5 digits is {1,3}.

Original entry on oeis.org

1, 3, 6, 8, 16, 18, 31, 33, 41, 43, 81, 83, 91, 93, 156, 158, 166, 168, 206, 208, 216, 218, 406, 408, 416, 418, 456, 458, 466, 468, 781, 783, 791, 793, 831, 833, 841, 843, 1031, 1033, 1041, 1043, 1081, 1083, 1091, 1093, 2031, 2033

Offset: 1

Clark Kimberling

Or, Numbers all of whose base-5 digits are odd.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A007091.

[n: n in [1..2500] | Set(IntegerToSequence(n, 5)) subset {1, 3}]; // Vincenzo Librandi, Jun 01 2012

Flatten[Table[FromDigits[#,5]&/@Tuples[{1,3},n],{n,5}]] (* Vincenzo Librandi, Jun 01 2012 *)

def A032912(n): return (int(bin(m:=n+1)[3:],5)<<1) + (5**(m.bit_length()-1)-1>>2) # Chai Wah Wu, Oct 13 2023

A032955 Numbers whose base-5 representation Sum_{i=0..m} d(i)*5^(m-i) has even d(i) for all odd i.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 10, 12, 14, 15, 17, 19, 20, 22, 24, 25, 26, 27, 28, 29, 35, 36, 37, 38, 39, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 60, 61, 62, 63, 64, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 85, 86, 87, 88, 89, 95, 96, 97, 98, 99, 100, 101

Offset: 1

Clark Kimberling

Sean A. Irvine, Table of n, a(n) for n = 1..10000

Cf. A007091 (numbers in base 5).

Definition corrected by Sean A. Irvine, Nov 16 2020

A058185 Numbers (written in decimal) which appear the same when written in base 5 and base 10/2.

Original entry on oeis.org

0, 1, 2, 3, 4, 10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 50, 51, 52, 53, 54, 60, 61, 62, 63, 64, 70, 71, 72, 73, 74, 100, 101, 102, 103, 104, 110, 111, 112, 113, 114, 120, 121, 122, 123, 124, 250, 251, 252, 253, 254, 260, 261, 262, 263, 264, 270, 271, 272, 273, 274

Offset: 1

Henry Bottomley, Nov 17 2000

To represent a number in base b, if a digit exceeds b-1, subtract b and carry 1. In fractional base b/c, subtract b and carry c. The sequence consists of numbers which in base 5 only have even digits, or one more than such numbers.

			10 is a term since it is written as 20 both in base 5 and base 10/2.
40 it not a term since it is written as 130 in base 5 and 80 in base 10/2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to fractional bases.

Cf. A007091, A024657, A058186.

s[n_] := s[n] = If[n == 0, 0, 10 * s[2 * Floor[n/10]] + Mod[n, 10]]; q[k_] := s[k] == FromDigits[IntegerDigits[k, 5]]; Select[Range[0, 300], q] (* Amiram Eldar, Aug 02 2025 *)

s(n) = if(n == 0, 0, 10 * s(n\10 * 2) + n % 10);
isok(k) = s(k) == fromdigits(digits(k, 5)); \\ Amiram Eldar, Aug 02 2025

Offset corrected by Amiram Eldar, Aug 02 2025

cp's OEIS Frontend

A024657 n written in fractional base 10/2.

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A037462 a(n) = Sum_{i = 0..m} d(i)*8^i, where Sum_{i = 0..m} d(i)*4^i is the base 4 representation of n.

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A083899 Number of divisors of n with largest digit <= 4 (base 10).

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A321882 a(n) is the least base b > 1 such that the sum n + n can be computed without carry.

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A004688 Fibonacci numbers written in base 5.

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A032543 Numbers that, when expressed in base 5 and then interpreted in base 10, yield a multiple of the original number.

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A032860 Numbers whose base-5 representation Sum_{i=0..m} d(i)*5^i has d(m) > d(m-1) < d(m-2) > ...

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A032912 Numbers whose set of base-5 digits is {1,3}.

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A037462 a(n) = Sum_{i = 0..m} d(i)8^i, where Sum_{i = 0..m} d(i)4^i is the base 4 representation of n.