A007091 -id:A007091 - cp's OEIS frontend

A033018 Numbers whose base-5 expansion has no run of digits with length < 2.

6, 12, 18, 24, 31, 62, 93, 124, 150, 156, 162, 168, 174, 300, 306, 312, 318, 324, 450, 456, 462, 468, 474, 600, 606, 612, 618, 624, 750, 775, 781, 787, 793, 799, 812, 843, 874, 1500, 1531, 1550, 1556, 1562, 1568, 1574, 1593

Offset: 1

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 1..1400

Cf. A007091. Supersequence of A033003.

Select[Range[10000],Min[Length/@Split[IntegerDigits[#, 5]]]>1&] (* Vincenzo Librandi, Feb 05 2014 *)

A039285 Numbers whose base-5 representation has the same nonzero number of 0's and 3's.

Original entry on oeis.org

15, 28, 40, 53, 65, 76, 77, 79, 80, 85, 95, 103, 115, 133, 138, 141, 142, 144, 148, 153, 165, 178, 190, 201, 202, 204, 205, 210, 220, 228, 240, 258, 263, 266, 267, 269, 273, 278, 290, 303, 315, 326, 327, 329, 330, 335, 345, 353, 365, 378, 381, 382, 384, 386

Offset: 1

Olivier Gérard

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

Cf. A007091.

Select[Range[400],DigitCount[#,5,0]==DigitCount[#,5,3]>0&] (* Harvey P. Dale, Apr 18 2014 *)

A039563 Numbers whose base-5 representation has the same number of 0's, 1's and 2's.

Original entry on oeis.org

3, 4, 18, 19, 23, 24, 27, 35, 51, 55, 93, 94, 98, 99, 118, 119, 123, 124, 138, 139, 142, 147, 178, 179, 190, 195, 202, 210, 227, 235, 258, 259, 266, 271, 278, 279, 290, 295, 326, 330, 351, 355, 382, 386, 402, 410, 426, 430, 468, 469, 473, 474, 493, 494, 498

Offset: 1

Olivier Gérard

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

Cf. A007091 (numbers in base 5).

f:=func; [k:k in [1..500]|f(k,5,0) eq f(k,5,1) and f(k,5,0) eq f(k,5,2)]; // Marius A. Burtea, Jan 17 2020

filter:= proc(n) local L,m;
  L:= sort(select(`<=`,convert(n,base,5),2));
  m:= nops(L)/3;
  m::integer and L = [0$m,1$m,2$m];
end proc:
select(filter, [$1..1000]); # Robert Israel, Jan 17 2020

Select[Range[500],DigitCount[#,5,0]==DigitCount[#,5,1] == DigitCount[ #,5,2]&] (* Harvey P. Dale, Dec 24 2021 *)

A102675 Number of digits >= 5 in decimal representation of n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Feb 03 2005

a(n) = 0 iff n is in A007091 (numbers in base 5). - Bernard Schott, Feb 02 2023

Curtis Cooper, Number of large digits in the positive integers not exceeding n, Abstracts Amer. Math. Soc., 25 (No. 1, 2004), p. 38, Abstract 993-11-964.

Hieronymus Fischer, Table of n, a(n) for n = 0..10000

Cf. A007091 (indices of 0's), A027868, A054899, A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564, A102676 (partial sums).

Cf. A000120, A000788, A023416, A059015 (for base 2).

p:=proc(n) local b,ct,j: b:=convert(n,base,10): ct:=0: for j from 1 to nops(b) do if b[j]>=5 then ct:=ct+1 else ct:=ct fi od: ct: end: seq(p(n),n=0..120); # Emeric Deutsch, Feb 23 2005

Table[Count[IntegerDigits[n],?(#>4&)],{n,0,120}] (* _Harvey P. Dale, Nov 13 2013 *)

From Hieronymus Fischer, Jun 10 2012: (Start)
a(n) = Sum_{j=1..m+1} (floor(n/10^j + 1/2) - floor(n/10^j)), where m = floor(log_10(n)).
G.f.: g(x) = (1/(1-x))*Sum_{j>=0} (x^(5*10^j) - x^(10*10^j))/(1 - x^10^(j+1)).
G.f.: g(x) = (1/(1-x))*Sum_{j>=0} x^(5*10^j)/(1 + x^(5*10^j)).  (End)

More terms from Emeric Deutsch, Feb 23 2005

A110595 a(1)=5. For n > 1, a(n) = 4*5^(n-1) = A005054(n).

Original entry on oeis.org

5, 20, 100, 500, 2500, 12500, 62500, 312500, 1562500, 7812500, 39062500, 195312500, 976562500, 4882812500, 24414062500, 122070312500, 610351562500, 3051757812500, 15258789062500, 76293945312500, 381469726562500

Offset: 1

Jonathan Vos Post, Jul 29 2005

a(n) is the number of n-digit integers that contain only even digits (A014263). - Bernard Schott, Nov 11 2022

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5).

Cf. A000351, A007091, A014263, A081604, A110591, A110592, A110593, A110594.

Join[{5},NestList[5#&,20,20]] (* Harvey P. Dale, Jun 19 2013 *)
Rest[CoefficientList[Series[5 x (1 - x)/(1 - 5 x), {x,0,50}], x]] (* G. C. Greubel, Sep 01 2017 *)

my(x='x+O('x^50)); Vec(5*x*(1-x)/(1-5*x)) \\ G. C. Greubel, Sep 01 2017

O.g.f.: 5*x*(1-x)/(1-5*x). - Better definition from R. J. Mathar, May 13 2008

Sum_{n>=1} 1/a(n) = 21/80. - Bernard Schott, Nov 11 2022

Better definition from R. J. Mathar, May 13 2008

Incorrect comment removed by Michel Marcus, Nov 11 2022

A137469 Numbers with an odd number of 1's in base 5 expansion.

Original entry on oeis.org

1, 5, 7, 8, 9, 11, 16, 21, 25, 27, 28, 29, 31, 35, 37, 38, 39, 40, 42, 43, 44, 45, 47, 48, 49, 51, 55, 57, 58, 59, 61, 66, 71, 76, 80, 82, 83, 84, 86, 91, 96, 101, 105, 107, 108, 109, 111, 116, 121, 125, 127, 128, 129, 131, 135, 137, 138, 139, 140, 142, 143, 144, 145

Offset: 1

Jonathan Vos Post, Apr 20 2008

This is to A000069 (Odious numbers: odd number of 1's in binary expansion) as A007091 (Numbers in base 5) is to A007088 (Numbers written in base 2). Note that odd number of 1's in base 3 expansion is simply the odd numbers.

			a(18) = 40 (base 10) is in the sequence because 40 (base 10) = 130 (base 5) which has an odd number (1) of 1's ; whereas 41 is not in the sequence because 41 (base 10) = 131 (base 5) has an even number (2) of 1's.

Index entries for 5-automatic sequences.

Cf. A000069, A007088, A007091.

isA137469 := proc(n) local a,d ; a := 0 ; for d in convert(n,base,5) do if d =1 then a := a+1 ; fi ; od; RETURN( a mod 2 = 1 ) ; end: for n from 1 to 400 do if isA137469(n) then printf("%d,",n) ; fi ; od: # R. J. Mathar, Apr 21 2008

Select[Range[200], OddQ[DigitCount[ #, 5][[1]]] &] (* Stefan Steinerberger, Apr 21 2008 *)

More terms from R. J. Mathar and Stefan Steinerberger, Apr 21 2008

A263611 Base 5 numbers whose square is a palindrome in base 5.

Original entry on oeis.org

0, 1, 2, 11, 101, 111, 231, 1001, 1111, 10001, 10101, 11011, 11204, 100001, 101101, 110011, 242204, 1000001, 1001001, 1010101, 1042214, 1100011, 2020303, 2043122, 2443304, 10000001, 10011001, 10100101, 11000011, 100000001, 100010001, 100101001, 101000101, 110000011, 111103411

Offset: 1

N. J. A. Sloane, Oct 23 2015

A029988 expressed in base 5.

G. J. Simmons, On palindromic squares of non-palindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 11-19. [Annotated scanned copy]

Cf. A007091, A002778, A029988, A263612, A029989.

With[{b = 5}, FromDigits@ IntegerDigits[#, b] & /@ Select[Range[b^9], PalindromeQ[IntegerDigits[#^2, b]] &]] (* Michael De Vlieger, Aug 15 2022 *)

a(n) = A007091(A029988(n)).

Name corrected by Charles R Greathouse IV, Aug 15 2022

A293292 Numbers with last digit less than 5 (in base 10).

Original entry on oeis.org

0, 1, 2, 3, 4, 10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 30, 31, 32, 33, 34, 40, 41, 42, 43, 44, 50, 51, 52, 53, 54, 60, 61, 62, 63, 64, 70, 71, 72, 73, 74, 80, 81, 82, 83, 84, 90, 91, 92, 93, 94, 100, 101, 102, 103, 104, 110, 111, 112, 113, 114, 120, 121, 122, 123, 124, 130

Offset: 1

Bruno Berselli, Oct 05 2017

Equivalently, numbers k such that floor(k/5) = 2*floor(k/10).
After 0, partial sums of A010122 starting from the 2nd term.
The sequence differs from A007091 after a(25).
Also numbers k such that floor(k/5) is even. - Peter Luschny, Oct 05 2017

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A010122, A239229, A257145, A293481 (complement).
Sequences of the type floor(n/d) = (10/d)*floor(n/10), where d is a factor of 10: A008592 (d=1), A197652 (d=2), this sequence (d=5), A001477 (d=10).
Sequences of the type n + r*floor(n/r): A005843 (r=1), A042948 (r=2), A047240 (r=3), A047476 (r=4), this sequence (r=5).

Magma
```
[n: n in [0..130] | n mod 10 lt 5];
```

[n: n in [0..130] | IsEven(Floor(n/5))];

Magma
```
[n+5*Floor(n/5): n in [0..70]];
```

select(k -> type(floor(k/5), even), [$0..130]); # Peter Luschny, Oct 05 2017

Table[n + 5 Floor[n/5], {n, 0, 70}]
Reap[For[k = 0, k <= 130, k++, If[Floor[k/5] == 2*Floor[k/10], Sow[k]]]][[2, 1]] (* or *) LinearRecurrence[{1, 0, 0, 0, 1, -1}, {0, 1, 2, 3, 4, 10}, 66] (* Jean-François Alcover, Oct 05 2017 *)

concat(0, Vec(x^2*(1 + x + x^2 + x^3 + 6*x^4) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)) + O(x^70))) \\ Colin Barker, Oct 05 2017

select(k->floor(k/5) == 2*floor(k/10), vector(1000, k, k)) \\ Colin Barker, Oct 05 2017

[k for k in range(131) if (k//5) % 2 == 0] # Peter Luschny, Oct 05 2017

def A293292(n): return (n-1<<1)-(n-1)%5 # Chai Wah Wu, Oct 29 2024

[k for k in (0..130) if 2.divides(floor(k/5))] # Peter Luschny, Oct 05 2017

G.f.: x^2*(1 + x + x^2 + x^3 + 6*x^4)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(n-1) + a(n-5) - a(n-6).
a(n) = (n-1) + 5*floor((n-1)/5) = 10*floor((n-1)/5) + ((n-1) mod 5).
a(n) = A257145(n+2) - A239229(n-1). - R. J. Mathar, Oct 05 2017
a(n) = 2n-2-((n-1) mod 5). - Chai Wah Wu, Oct 29 2024

Definition by David A. Corneth, Oct 05 2017

A331561 The base-10 numbers with a digit product > 0 and which when written in bases 3,4,5,6,7,8,9 have three or more other base representations with the same digit product.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 12563124891, 115233863842, 123858133813, 363254652118, 1324658354423, 1511864334458, 1825693128524, 2321856215149, 2632654133853, 3146626254542, 3521445321466, 12462982162122, 12496523158865, 13129883155583, 13443165514365, 14213435966581

Offset: 1

Scott R. Shannon, Jan 20 2020

This is a subsequence of A331565.

			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.
12563124891 is a term as 12563124891_10 = 5434343211123_6 = 623216541162_7 = 135464411233_8, so it has three other base representations where the digit product also equals 103680.
115233863842 is a term as 115233863842_10 = 124534313342214_6 = 11216413452466_7 = 1532436234242_8, so it has three other base representations where the digit product also equals 829440.

Giovanni Resta, Table of n, a(n) for n = 1..61 (terms < 10^14)

Cf. A007954, A007089, A007090, A007091, A007092, A007093, A007094, A007095, A331565.

Terms a(10) and beyond from Giovanni Resta, Jan 27 2020

A374735 a(n) is the least k > 0 such that n and k*n can be added without carries in decimal.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 5, 10, 1, 1, 1, 1, 1, 2, 2, 3, 10, 20, 1, 1, 1, 1, 1, 2, 2, 6, 5, 30, 1, 1, 1, 1, 1, 4, 7, 3, 20, 40, 1, 1, 1, 1, 1, 10, 5, 3, 5, 50, 2, 2, 2, 2, 6, 2, 2, 6, 30, 60, 2, 2, 2, 2, 5, 2, 2, 3, 15, 70, 3, 3, 3, 7, 3, 4, 12, 13, 40, 80, 5, 5

Offset: 0

Rémy Sigrist, Jul 18 2024

			For n = 8:
- 1*8 = 8; computing 8 + 8 requires a carry,
- 2*8 = 16; computing 8 + 16 requires a carry,
- 3*8 = 24; computing 8 + 24 requires a carry,
- 4*8 = 32; computing 8 + 32 requires a carry,
- 5*8 = 40; computing 8 + 40 does not require a carry,
- so a(8) = 5.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Cf. A007091, A261891 (analog for binary), A353623 (analog for balanced ternary), A374736.

a(n, base = 10) = { for (k = 1, oo, if (sumdigits((k+1)*n, base) == sumdigits(n, base) + sumdigits(k*n, base), return (k););); }

from itertools import count
def A374735(n):
    s = list(map(int,str(n)[::-1]))
    return next(k for k in count(1) if all(a+b<=9 for a, b in zip(s,map(int,str(k*n)[::-1])))) # Chai Wah Wu, Jul 19 2024

a(n) = 1 iff n belongs to A007091.

a(10*n) = a(n).

cp's OEIS Frontend

A033018 Numbers whose base-5 expansion has no run of digits with length < 2.

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A039285 Numbers whose base-5 representation has the same nonzero number of 0's and 3's.

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A039563 Numbers whose base-5 representation has the same number of 0's, 1's and 2's.

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A102675 Number of digits >= 5 in decimal representation of n.

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A110595 a(1)=5. For n > 1, a(n) = 4*5^(n-1) = A005054(n).

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A137469 Numbers with an odd number of 1's in base 5 expansion.

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A263611 Base 5 numbers whose square is a palindrome in base 5.

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A293292 Numbers with last digit less than 5 (in base 10).

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