A007091 -id:A007091 - cp's OEIS frontend

A058186 Numbers (written in base 5) which appear the same when written in base 5 and base 10/2.

0, 1, 2, 3, 4, 20, 21, 22, 23, 24, 40, 41, 42, 43, 44, 200, 201, 202, 203, 204, 220, 221, 222, 223, 224, 240, 241, 242, 243, 244, 400, 401, 402, 403, 404, 420, 421, 422, 423, 424, 440, 441, 442, 443, 444, 2000, 2001, 2002, 2003, 2004, 2020, 2021, 2022, 2023

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 have digits in {0,2,4} except that the unit digit can be any from {0,1,2,3,4}.

			20 (10 in decimal) is a term since it is written as 20 both in base 5 and base 10/2.
30 (15 in decimal) is not a term since it is written as 30 in base 5 and 25 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, A058185.

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

s(n) = if(n == 0, 0, 10 * s(n\10 * 2) + n % 10);
f(n) = fromdigits(digits(n, 5));
list(lim) = apply(f, select(x -> s(x) == f(x), vector(lim+1, i, i-1))); \\ Amiram Eldar, Aug 02 2025

a(n) = A007091(A058185(n)). - Amiram Eldar, Aug 02 2025

Offset corrected by Amiram Eldar, Aug 02 2025

A067044 Smallest positive k such that k*n contains only even digits.

Original entry on oeis.org

2, 1, 2, 1, 4, 1, 4, 1, 32, 2, 2, 2, 2, 2, 4, 3, 4, 16, 12, 1, 2, 1, 2, 1, 8, 1, 18, 1, 14, 2, 2, 2, 2, 2, 8, 8, 6, 6, 12, 1, 2, 1, 2, 1, 64, 1, 6, 1, 14, 4, 4, 4, 8, 9, 4, 4, 4, 7, 14, 1, 4, 1, 14, 1, 4, 1, 4, 1, 12, 4, 4, 4, 28, 3, 8, 3, 6, 6, 34, 1, 6, 1, 8, 1, 8, 1, 24, 1, 32, 32, 22, 5, 22, 3

Offset: 1

Amarnath Murthy, Dec 29 2001

No multiple of 10 can appear in this sequence. - M. F. Hasler, Mar 07 2025

			a(7) = 4 as among the multiples of 7 (i.e., 7, 14, 21, 28...), 28 is the smallest multiple with only even digits and a(7)= 28/7 = 4.
a(16) = 3 is the first odd term > 1, a(n = 54, 58, 74, 76, 92, 94, 96, 98, ...) are the next examples, cf. A380874. - _M. F. Hasler_, Mar 03 2025

Paul Tek, Table of n, a(n) for n = 1..10000

Cf. A061807, A380874.

Cf. A014263 (numbers with only even digits), A007091 (numbers in base 5).

Table[k = n; While[Length[Intersection[{1, 3, 5, 7, 9}, IntegerDigits[k]]] > 0, k = k + n]; k/n, {n, 100}] (* T. D. Noe, Jun 03 2013 *)
sk[n_]:=Module[{k=1},While[!AllTrue[IntegerDigits[k*n],EvenQ],k++];k]; Array[sk,100] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 27 2015 *)

apply( {A067044(n, f=1+n%2)=forstep(a=f*n, oo, f*n, digits(a)%2||return(a/n))}, [1..99]) \\ M. F. Hasler, Mar 03 2025

A067044 = lambda n: next(k for k in range(1+n%2, 9<<99, 1+n%2)if not any(int(d)&1 for d in str(n*k))) # M. F. Hasler, Mar 03 2025

From M. F. Hasler, Mar 07 2025: (Start)
There is an explicit formula for many values of n:
a(n) = 1 if n has only even digits <=> n is in A014263, else:
a(n) = 2 if n has only digits < 5 <=> n is in A007091;
a(m*(10^k-1)) = 8*round(10^k/6)^2/m for m = 1, 2, 4 or 8 and any k > 0;
a(5*(10^k-1)) = 16*round(10^k/6)^2 for any k > 0;
a(50*m + {5 or 15}) = 4 if m has all digits < 5. (End)

More terms from Eli McGowan (ejmcgowa(AT)mail.lakeheadu.ca), May 06 2002

Data corrected by Paul Tek, Jun 03 2013

A072806 Primes of the form 6k+5 written in base 5.

Original entry on oeis.org

10, 21, 32, 43, 104, 131, 142, 203, 214, 241, 313, 324, 401, 412, 423, 1011, 1022, 1044, 1132, 1143, 1204, 1231, 1242, 1402, 1413, 1424, 2001, 2012, 2023, 2034, 2111, 2133, 2221, 2232, 2342, 2403, 2414, 3013, 3024, 3101, 3134, 3211, 3233, 3244, 3321

Offset: 1

Labos Elemer, Jul 12 2002

			41 = 25 + 3*5 + 1 = 131_5.

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

Cf. A007091, A007528.

Cf. A008713, A072803, A072804, A072805, A072807.

Do[s=Prime[n]; If[Mod[s, 6]==5, Print[BaseForm[s, 5]]], {n, 1, 256}]
FromDigits[IntegerDigits[#, 5]] & /@  Select[Table[6 n + 5, {n, 0, 100}], PrimeQ] (* Harvey P. Dale, Oct 05 2023 *)

lista(nn) = for (n=0, nn, if (isprime(p=6*n+5), print1(fromdigits(digits(p, 5)), ", "))); \\ Michel Marcus, Jul 09 2018

a(n) = A007091(A007528(n)). - Michel Marcus, Jul 09 2018

A158082 Squares whose decimal expansion contains no digit greater than 4.

Original entry on oeis.org

0, 1, 4, 100, 121, 144, 324, 400, 441, 1024, 1444, 2304, 2401, 10000, 10201, 10404, 12100, 12321, 14400, 22201, 23104, 32041, 32400, 33124, 40000, 40401, 44100, 101124, 102400, 103041, 110224, 114244, 121104, 123201, 130321, 131044, 144400, 203401, 204304, 213444

Offset: 1

N. J. A. Sloane, May 23 2010

Erich Friedman, What's Special About This Number?

Cf. A158304.

Intersection of A000290 and A007091.

Select[Range[0, 1000]^2, Max[IntegerDigits[#]] <= 4 &] (* Paolo Xausa, Apr 29 2024 *)

A255590 Convert n to base 5, move the least significant digit to the most significant one and convert back to base 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 1, 6, 11, 16, 21, 2, 7, 12, 17, 22, 3, 8, 13, 18, 23, 4, 9, 14, 19, 24, 5, 30, 55, 80, 105, 6, 31, 56, 81, 106, 7, 32, 57, 82, 107, 8, 33, 58, 83, 108, 9, 34, 59, 84, 109, 10, 35, 60, 85, 110, 11, 36, 61, 86, 111, 12, 37, 62, 87, 112, 13, 38, 63, 88

Offset: 0

Paolo P. Lava, Mar 02 2015

a(5*n) = n.
a(5^n) = 5^(n-1).
Fixed points of the transform are listed in A048330.

			14 in base 5 is 24: moving the least significant digit to the most significant one we have 42 that is 22 in base 10.

Indranil Ghosh, Table of n, a(n) for n = 0..15625 (terms 0..1000 from Paolo P. Lava)

Cf. A030104, A038572, A048330, A255588, A255589, A255591-A255594, A048787, A255689-A255693.

with(numtheory): P:=proc(q,h) local a,b,k,n; print(0);
for n from 1 to q do
a:=convert(n,base,h); b:=[]; for k from 2 to nops(a) do b:=[op(b),a[k]]; od; a:=[op(b),a[1]];
a:=convert(a,base,h,10); b:=0; for k from nops(a) by -1 to 1 do b:=10*b+a[k]; od;
print(b); od; end: P(10^4,5);

roll[n_, b_] := Block[{w = IntegerDigits[n, b]}, Prepend[Most@ w, Last@ w]]; b = 5; FromDigits[#, b] & /@ (roll[#, b] & /@ Range[0, 68]) (* Michael De Vlieger, Mar 04 2015 *)
Table[FromDigits[RotateRight[IntegerDigits[n,5]],5],{n,0,100}] (* Harvey P. Dale, Jun 11 2025 *)

def A255590(n):
    x=str(A007091(n))
    return int(x[-1]+x[:-1], 5) # Indranil Ghosh, Feb 03 2017

A331565 The base 10 numbers with a digit product > 0 and which when written in bases 3,4,5,6,7,8,9 have two or more other base representations with the same digit product.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 91, 491, 921, 1138, 1234, 4853, 13581, 23568, 29242, 42161, 42162, 42163, 42164, 42991, 43365, 44313, 83342, 83651, 85226, 114382, 153881, 155462, 159422, 232868, 291862, 296183, 352486, 372642, 398543, 419563, 441194, 465326, 616146, 625431, 625523, 635813

Offset: 1

Scott R. Shannon, Jan 20 2020

For terms 10 < a(n) < 10^9 none have a base-3 representation whose digit product equals the base-10 product. The first such entry using the base-4 representation is 491.

			6 is a term as 6_10 = 6_7 = 6_8 = 6_9, so it has three other base representations where the digit product also equals 6.
91 is a term as 91_10 = 331_5 = 133_8, so it has two other base representations where the digit product also equals 9.
491 is a term as 491_10 = 13223_4 = 3431_5, so it has two other base representations where the digit product also equals 36.

Cf. A007954, A007089, A007090, A007091, A007092, A007093, A007094, A007095, A331561.

Subsequence of A052382 (zeroless numbers).

proDig[n_, b_] := Times @@ IntegerDigits[n, b]; seqQ[n_] := Module[{prod = proDig[n, 10], count = 0}, If[prod > 0, Do[If[proDig[n, b] == prod, count++]; If[count == 2, Break[]], {b, 3, 9}]]; count == 2]; Select[Range[650000], seqQ] (* Amiram Eldar, Jan 21 2020 *)

isok(n) = {my(p=vecprod(digits(n))); (p != 0) && (sum(k=3, 9, p==vecprod(digits(n,k))) >= 2);} \\ Michel Marcus, Jan 21 2020

A339255 Leading digit of n in base 5.

Original entry on oeis.org

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

Offset: 1

Kevin Ryde, Nov 28 2020

Kevin Ryde, Table of n, a(n) for n = 1..10000

Cf. A007091 (base 5), A073851 (partial sums).

In other bases: A122586, A122587, A339256, A000030, A099563, A276153.

IntegerDigits[#,5][[1]]&/@Range[100] (* Harvey P. Dale, Sep 04 2021 *)

PARI
```
a(n) = n\5^logint(n,5);
```

a(n) = floor(n / 5^floor(log_5(n))).

G.f.: (x + Sum_{k>=0} Sum_{d=2..4} (x^(d*5^k)-x^(5^(k+1))) )/(1-x).

A031950 Numbers with exactly two distinct base-5 digits.

Original entry on oeis.org

5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 26, 30, 32, 33, 34, 36, 37, 41, 43, 46, 49, 50, 52, 56, 57, 60, 61, 63, 64, 67, 68, 72, 74, 75, 78, 81, 83, 87, 88, 90, 91, 92, 94, 98, 99, 100, 104, 106, 109, 112, 114, 118, 119

Offset: 1

Clark Kimberling

Index entries for 5-automatic sequences.

Cf. A007091.

Select[Range[150],Length[Union[IntegerDigits[#,5]]]==2&]  (* Harvey P. Dale, Apr 13 2011 *)

A032542 Integer part of decimal 'base-5 looking' numbers divided by their actual base-5 values.

Original entry on oeis.org

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

Offset: 1

Patrick De Geest, Apr 15 1998

Cf. A032543, A032544. See also A032532 for explanation.

a(n) = floor(A007091(n)/n). - Sean A. Irvine, Jun 22 2020

Offset and data corrected by Sean A. Irvine, Jun 22 2020

A032829 Numbers whose set of base-5 digits is {3,4}.

Original entry on oeis.org

3, 4, 18, 19, 23, 24, 93, 94, 98, 99, 118, 119, 123, 124, 468, 469, 473, 474, 493, 494, 498, 499, 593, 594, 598, 599, 618, 619, 623, 624, 2343, 2344, 2348, 2349, 2368, 2369, 2373, 2374, 2468, 2469, 2473, 2474, 2493, 2494, 2498

Offset: 1

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for 5-automatic sequences.

Cf. A007091.

[n: n in [1..2800] | Set(IntegerToSequence(n, 5)) subset {3, 4}];// Vincenzo Librandi, May 29 2012

Flatten[Table[FromDigits[#,5]&/@Tuples[{3,4},n],{n,5}]] (* Vincenzo Librandi, May 29 2012 *)

a[1]:3$ a[2]:4$ a[n]:= if oddp(n) then 5*a[floor(n/2)]+3 else 5*a[floor((n-1)/2)]+4$ makelist(a[n],n,1,45); /* Bruno Berselli, May 30 2012 */

a(1)=3, a(2)=4; a(n) = 5*a(floor(n/2))+3 for n odd, otherwise a(n) = 5*a(floor((n-1)/2))+4. - Bruno Berselli, May 30 2012

cp's OEIS Frontend

A058186 Numbers (written in base 5) which appear the same when written in base 5 and base 10/2.

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A067044 Smallest positive k such that k*n contains only even digits.

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A072806 Primes of the form 6k+5 written in base 5.

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A158082 Squares whose decimal expansion contains no digit greater than 4.

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A255590 Convert n to base 5, move the least significant digit to the most significant one and convert back to base 10.

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A331565 The base 10 numbers with a digit product > 0 and which when written in bases 3,4,5,6,7,8,9 have two or more other base representations with the same digit product.

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A339255 Leading digit of n in base 5.

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A031950 Numbers with exactly two distinct base-5 digits.