cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 12 results. Next

A007953 Digital sum (i.e., sum of digits) of n; also called digsum(n).

Original entry on oeis.org

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

Views

Author

R. Muller

Keywords

Comments

Do not confuse with the digital root of n, A010888 (first term that differs is a(19)).
Also the fixed point of the morphism 0 -> {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, 1 -> {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, 2 -> {2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, etc. - Robert G. Wilson v, Jul 27 2006
For n < 100 equal to (floor(n/10) + n mod 10) = A076314(n). - Hieronymus Fischer, Jun 17 2007
It appears that a(n) is the position of 10*n in the ordered set of numbers obtained by inserting/placing one digit anywhere in the digits of n (except a zero before 1st digit). For instance, for n=2, the resulting set is (12, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 32, 42, 52, 62, 72, 82, 92) where 20 is at position 2, so a(2) = 2. - Michel Marcus, Aug 01 2022
Also the total number of beads required to represent n on a Russian abacus (schoty). - P. Christopher Staecker, Mar 31 2023
a(n) / a(2n) <= 5 with equality iff n is in A169964, while a(n) / a(3n) is unbounded, since if n = (10^k + 2)/3, then a(n) = 3*k+1, a(3n) = 3, so a(n) / a(3n) = k + 1/3 -> oo when k->oo (see Diophante link). - Bernard Schott, Apr 29 2023
Also the number of symbols needed to write number n in Egyptian numerals for n < 10^7. - Wojciech Graj, Jul 10 2025

Examples

			a(123) = 1 + 2 + 3 = 6, a(9875) = 9 + 8 + 7 + 5 = 29.

Links

Crossrefs

Cf. A003132, A055012, A055013, A055014, A055015, A010888, A007954, A031347, A055017, A076313, A076314, A054899, A138470, A138471, A138472, A000120, A004426, A004427, A054683, A054684, A069877, A179082-A179085, A108971, A169964, A179987, A179988, A180018, A180019, A217928, A216407, A037123, A074784, A231688, A231689, A225693, A254524 (ordinal transform).
Bisections: A004092, A004155.
For n + digsum(n) see A062028.

Programs

  • Haskell
    a007953 n | n < 10 = n
          | otherwise = a007953 n' + r where (n',r) = divMod n 10
-- Reinhard Zumkeller, Nov 04 2011, Mar 19 2011
  • Magma
    [ &+Intseq(n): n in [0..87] ];  // Bruno Berselli, May 26 2011
  • Maple
    A007953 := proc(n) add(d,d=convert(n,base,10)) ; end proc: # R. J. Mathar, Mar 17 2011
  • Mathematica
    Table[Sum[DigitCount[n][[i]] * i, {i, 9}], {n, 50}] (* Stefan Steinerberger, Mar 24 2006 *)
Table[Plus @@ IntegerDigits @ n, {n, 0, 87}] (* or *)
Nest[Flatten[# /. a_Integer -> Array[a + # &, 10, 0]] &, {0}, 2] (* Robert G. Wilson v, Jul 27 2006 *)
Total/@IntegerDigits[Range[0,90]] (* Harvey P. Dale, May 10 2016 *)
DigitSum[Range[0, 100]] (* Requires v. 14 *) (* Paolo Xausa, May 17 2024 *)
  • PARI
    a(n)=if(n<1, 0, if(n%10, a(n-1)+1, a(n/10))) \\ Recursive, very inefficient. A more efficient recursive variant: a(n)=if(n>9, n=divrem(n, 10); n[2]+a(n[1]), n)
  • PARI
    a(n, b=10)={my(s=(n=divrem(n, b))[2]); while(n[1]>=b, s+=(n=divrem(n[1], b))[2]); s+n[1]} \\ M. F. Hasler, Mar 22 2011
  • PARI
    a(n)=sum(i=1, #n=digits(n), n[i]) \\ Twice as fast. Not so nice but faster:
  • PARI
    a(n)=sum(i=1,#n=Vecsmall(Str(n)),n[i])-48*#n \\ M. F. Hasler, May 10 2015
/* Since PARI 2.7, one can also use: a(n)=vecsum(digits(n)), or better: A007953=sumdigits. [Edited and commented by M. F. Hasler, Nov 09 2018] */
  • PARI
    a(n) = sumdigits(n); \\ Altug Alkan, Apr 19 2018
  • Python
    def A007953(n):
    return sum(int(d) for d in str(n)) # Chai Wah Wu, Sep 03 2014
  • Python
    def a(n): return sum(map(int, str(n))) # Michael S. Branicky, May 22 2021
  • Scala
    (0 to 99).map(.toString.map(.toInt - 48).sum) // Alonso del Arte, Sep 15 2019
  • Smalltalk
    "Recursive version for general bases. Set base = 10 for this sequence."
digitalSum: base
| s |
base = 1 ifTrue: [^self].
(s := self // base) > 0
  ifTrue: [^(s digitalSum: base) + self - (s * base)]
  ifFalse: [^self]
"by Hieronymus Fischer, Mar 24 2014"
  • Swift
    A007953(n): String(n).compactMap{$0.wholeNumberValue}.reduce(0, +) // Egor Khmara, Jun 15 2021

Formula

a(A051885(n)) = n.
a(n) <= 9(log_10(n)+1). - Stefan Steinerberger, Mar 24 2006
From Benoit Cloitre, Dec 19 2002: (Start)
a(0) = 0, a(10n+i) = a(n) + i for 0 <= i <= 9.
a(n) = n - 9*(Sum_{k > 0} floor(n/10^k)) = n - 9*A054899(n). (End)
From Hieronymus Fischer, Jun 17 2007: (Start)
G.f. g(x) = Sum_{k > 0, (x^k - x^(k+10^k) - 9x^(10^k))/(1-x^(10^k))}/(1-x).
a(n) = n - 9*Sum_{10 <= k <= n} Sum_{j|k, j >= 10} floor(log_10(j)) - floor(log_10(j-1)). (End)
From Hieronymus Fischer, Jun 25 2007: (Start)
The g.f. can be expressed in terms of a Lambert series, in that g(x) = (x/(1-x) - 9*L[b(k)](x))/(1-x) where L[b(k)](x) = sum{k >= 0, b(k)*x^k/(1-x^k)} is a Lambert series with b(k) = 1, if k > 1 is a power of 10, else b(k) = 0.
G.f.: g(x) = (Sum_{k > 0} (1 - 9*c(k))*x^k)/(1-x), where c(k) = Sum_{j > 1, j|k} floor(log_10(j)) - floor(log_10(j-1)).
a(n) = n - 9*Sum_{0 < k <= floor(log_10(n))} a(floor(n/10^k))*10^(k-1). (End)
From Hieronymus Fischer, Oct 06 2007: (Start)
a(n) <= 9*(1 + floor(log_10(n))), equality holds for n = 10^m - 1, m > 0.
lim sup (a(n) - 9*log_10(n)) = 0 for n -> oo.
lim inf (a(n+1) - a(n) + 9*log_10(n)) = 1 for n -> oo. (End)
a(n) = A138530(n, 10) for n > 9. - Reinhard Zumkeller, Mar 26 2008
a(A058369(n)) = A004159(A058369(n)); a(A000290(n)) = A004159(n). - Reinhard Zumkeller, Apr 25 2009
a(n) mod 2 = A179081(n). - Reinhard Zumkeller, Jun 28 2010
a(n) <= 9*log_10(n+1). - Vladimir Shevelev, Jun 01 2011
a(n) = a(n-1) + a(n-10) - a(n-11), for n < 100. - Alexander R. Povolotsky, Oct 09 2011
a(n) = Sum_{k >= 0} A031298(n, k). - Philippe Deléham, Oct 21 2011
a(n) = a(n mod b^k) + a(floor(n/b^k)), for all k >= 0. - Hieronymus Fischer, Mar 24 2014
Sum_{n>=1} a(n)/(n*(n+1)) = 10*log(10)/9 (Shallit, 1984). - Amiram Eldar, Jun 03 2021

Extensions

More terms from Hieronymus Fischer, Jun 17 2007
Edited by Michel Marcus, Nov 11 2013

A054683 Numbers whose sum of digits is even.

Original entry on oeis.org

0, 2, 4, 6, 8, 11, 13, 15, 17, 19, 20, 22, 24, 26, 28, 31, 33, 35, 37, 39, 40, 42, 44, 46, 48, 51, 53, 55, 57, 59, 60, 62, 64, 66, 68, 71, 73, 75, 77, 79, 80, 82, 84, 86, 88, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 110, 112, 114, 116, 118, 121, 123, 125, 127, 129, 130
Offset: 1

Views

Author

Odimar Fabeny, Apr 19 2000

Keywords

Comments

Union of A179082 and A179084; A179081(a(n)) = 0. - Reinhard Zumkeller, Jun 28 2010
Integers with an even number of odd digits. - Bernard Schott, Nov 18 2022

Examples

			0, 2, 4, 6, 8, 11 (2), 13 (4), 15 (6), 17 (8), 19 (10), 20 (2), 22 (4) and so on.

Links

Crossrefs

Cf. A179081, A270264.
Subsequences: A014263, A099814, A179082, A179084.
Similar: A054684 (with an odd number of odd digits), A356929 (with an even number of even digits).

Programs

  • Mathematica
    Select[Range[0,200],EvenQ[Total[IntegerDigits[#]]]&] (* Harvey P. Dale, Jan 04 2015 *)
  • PARI
    is(n)=my(d=digits(n));sum(i=1,#d,d[i])%2==0 \\ Charles R Greathouse IV, Aug 09 2013
  • PARI
    a(n) = n--; m = 10*(n\5); s=sumdigits(m); m + (1-(s-1)%2) + 2*(n%5) \\ David A. Corneth, Jun 05 2016
  • Python
    A054683_list = [i for i in range(10**3) if not sum(int(d) for d in str(i)) % 2] # Chai Wah Wu, Mar 17 2016

Formula

a(n) = 2*n for the first 5 terms; a(n) = 2*n + 1 for the next 5 terms (recurrence).
I.e., for n > 0, a(n + 10) = a(n) + 20. - David A. Corneth, Jun 05 2016

Extensions

More terms from James Sellers, Apr 19 2000
Example corrected by David A. Corneth, Jun 05 2016

A179081 Parity of sum of digits of n.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1
Offset: 0

Views

Author

Reinhard Zumkeller, Jun 28 2010

Keywords

Comments

a(n) = A000035(A007953(n));
characteristic function of numbers with an odd sum of digits in their decimal representation:
a(A054684(n)) = 1; a(A054683(n)) = 0;
a(A179083(n)) = a(A179085(n)) = 1;
a(A179082(n)) = a(A179084(n)) = 0.

Examples

			a(789) = (7+8+9) mod 2 = 0;
a(790) = (7+9+0) mod 2 = 0;
a(791) = (7+9+1) mod 2 = 1.

Links

Programs

  • Mathematica
    Array[Mod[Total[IntegerDigits[#]],2]&,110,0] (* Harvey P. Dale, Feb 20 2013 *)
  • PARI
    a(n) = vecsum(digits(n)) % 2 \\ Jeppe Stig Nielsen, Jan 06 2018

Formula

a(n) = if n<10 then (n mod 2) else a([n/10]) XOR (n mod 2).

A179085 Odd numbers having an odd sum of digits in their decimal representation.

Original entry on oeis.org

1, 3, 5, 7, 9, 21, 23, 25, 27, 29, 41, 43, 45, 47, 49, 61, 63, 65, 67, 69, 81, 83, 85, 87, 89, 111, 113, 115, 117, 119, 131, 133, 135, 137, 139, 151, 153, 155, 157, 159, 171, 173, 175, 177, 179, 191, 193, 195, 197, 199, 201, 203, 205, 207, 209, 221, 223, 225, 227
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 28 2010

Keywords

Comments

a(n) = A030142(n) for n <= 25;
intersection of A005408 and A054684: A000035(a(n))*A179081(a(n))=1;
complement of A179084 with respect to A005408;
complement of A179083 with respect to A054684;
a(n) mod 2 = 1 and A007953(a(n)) mod 2 = 1.

Links

Programs

  • Mathematica
    Select[Range[1,227,2], OddQ[Total[IntegerDigits[#]]] &] (* Jayanta Basu, May 07 2013 *)

A179083 Even numbers having an odd sum of digits in their decimal representation.

Original entry on oeis.org

10, 12, 14, 16, 18, 30, 32, 34, 36, 38, 50, 52, 54, 56, 58, 70, 72, 74, 76, 78, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 120, 122, 124, 126, 128, 140, 142, 144, 146, 148, 160, 162, 164, 166, 168, 180, 182, 184, 186, 188, 210, 212, 214, 216, 218, 230, 232, 234
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 28 2010

Keywords

Comments

a(n) = A007958(n) for n <= 30;
intersection of A005843 and A054684: A059841(a(n))*A179081(a(n))=1;
complement of A179082 with respect to A005843;
complement of A179085 with respect to A054684;
a(n) mod 2 = 0 and A007953(a(n)) mod 2 = 1.

Links

A137233 Number of n-digit even numbers.

Original entry on oeis.org

5, 45, 450, 4500, 45000, 450000, 4500000, 45000000, 450000000, 4500000000, 45000000000, 450000000000, 4500000000000, 45000000000000, 450000000000000, 4500000000000000, 45000000000000000, 450000000000000000, 4500000000000000000, 45000000000000000000, 450000000000000000000
Offset: 1

Views

Author

Ctibor O. Zizka, Mar 08 2008

Keywords

Comments

From Kival Ngaokrajang, Oct 18 2013: (Start)
a(n) is also the total number of double rows identified numbers in n digit.
For example:
n = 1: 01 23 45 67 89 = 5 double rows;
n = 2: 1011 1213 1415 1617 1819...9899 = 45 double rows;
n = 3: 100101 102103 104105...998999 = 450 double rows;
The number of double rows is also A030656. (End)
a(n) is also the number of n-digit integers with an even number of even digits (A356929); a(5) = 45000 is the answer to the question 2 of the Olympiade Mathématique Belge in 2004 (link). - Bernard Schott, Sep 06 2022
a(n) is also the number of n-digit integers with an odd number of odd digits (A054684). - Bernard Schott, Nov 07 2022

Examples

			a(2) = 45 because there are 45 2-digit even numbers.

Links

Crossrefs

Cf. A000422, A002275, A002276, A011577, A014923, A014925, A016313, A019518, A037487, A053052, A057138, A090843, A097166, A099914, A099915.
Cf. A030656, A052268, A054684, A356929.

Programs

Formula

a(n) = 9*10^(n-1)/2 if n > 1. - R. J. Mathar, May 23 2008
From Elmo R. Oliveira, Jul 23 2025: (Start)
G.f.: 5*x*(1-x)/(1-10*x).
E.g.f.: (-9 + 10*x + 9*exp(10*x))/20.
a(n) = 10*a(n-1) for n > 2.
a(n) = A052268(n)/2 for n >= 2. (End)

Extensions

Corrected and extended by R. J. Mathar, May 23 2008
More terms from Elmo R. Oliveira, Jul 23 2025

A318700 Positive numbers that contain odd and even digits.

Original entry on oeis.org

10, 12, 14, 16, 18, 21, 23, 25, 27, 29, 30, 32, 34, 36, 38, 41, 43, 45, 47, 49, 50, 52, 54, 56, 58, 61, 63, 65, 67, 69, 70, 72, 74, 76, 78, 81, 83, 85, 87, 89, 90, 92, 94, 96, 98, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 112, 114, 116, 118, 120, 121, 122
Offset: 1

Views

Author

Enrique Navarrete, Aug 31 2018

Keywords

Comments

The sequence of first differences takes on the values {1, 2, 3} only, and each of these values occurs infinitely often (the values 1 and 2 are clear; for the value 3, note that consecutive numbers such as 199..9, 200..0 and 399..9, 400..0 that are excluded from the sequence occur infinitely often).
Numbers n such that A065031(n) is a term of A111066. - Felix Fröhlich, Sep 01 2018
Nonnegative integers excluding those such that digits in their decimal representation share all the same parity. - R. J. Cano, Sep 10 2018

Examples

			49 and 50 are in the sequence but 19 and 20 are not.

Links

Crossrefs

Cf. A065031, A111066, A030141, A054684, A227870.

Programs

  • Mathematica
    Select[Range@ 122, Union[Mod[IntegerDigits[#], 2]] == {0, 1} &] (* Michael De Vlieger, Sep 04 2018 *)
  • PARI
    is(n) = my(d=digits(n), v=[]); if(n < 10, return(0)); for(k=1, #d, v=concat(v, [d[k]%2])); vecmin(v)!=vecmax(v) \\ Felix Fröhlich, Sep 01 2018
  • PARI
    See Cano link.

A118732 Numbers k such that 3^k has odd digit sum.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 9, 10, 11, 13, 14, 16, 17, 18, 20, 21, 22, 25, 26, 27, 30, 31, 32, 34, 35, 36, 39, 41, 45, 48, 51, 52, 53, 59, 60, 61, 62, 63, 65, 66, 68, 69, 73, 74, 76, 78, 79, 80, 81, 86, 87, 89, 91, 92, 98, 99, 100, 101, 103, 105, 108, 113, 114, 115, 117, 118, 119, 121
Offset: 1

Views

Author

Zak Seidov, May 22 2006

Keywords

Comments

Numbers k such that A000244(k) is in A054684. - Robert Israel, Jun 28 2017

Links

Crossrefs

Cf. A000244, A054684, A118731, A118735.

Programs

  • Maple
    select(t -> convert(convert(3^t,base,10),`+`)::odd, [$0..1000]); # Robert Israel, Jun 28 2017
  • Mathematica
    Select[Range[0, 121], Mod[ Plus @@ IntegerDigits[3^# ], 2] == 1 &] (* Ray Chandler, Jun 10 2006 *)
Select[Range[0,200],OddQ[Total[IntegerDigits[3^#]]]&] (* Harvey P. Dale, Dec 30 2021 *)
  • PARI
    isok(n) = (sumdigits(3^n) % 2) == 1; \\ Michel Marcus, Jun 28 2017

A064507 Numbers of more than one digit that yield primes when cast in their own base.

Original entry on oeis.org

21, 27, 29, 61, 67, 111, 151, 191, 201, 203, 223, 241, 313, 319, 331, 351, 373, 397, 403, 409, 461, 463, 481, 553, 559, 571, 667, 711, 791, 807, 841, 869, 889, 931, 979, 1071, 1079, 1107, 1129, 1189, 1257, 1273, 1277, 1297, 1431, 1437, 1583, 1611, 1639
Offset: 1

Views

Author

Jon Perry, Oct 06 2001

Keywords

Comments

All terms are coprime to 10. - Robert Israel, Dec 04 2016

Examples

			E.g. 21 in base 21 is 2*21+1=43, 111 is 1*111*111+1*111+1=12433, etc.

Links

Crossrefs

Cf. A064508. Contained in A054684.

Programs

  • Maple
    filter:= proc(n) local L,i;
   L:= convert(n,base, 10);
   isprime(add(L[i]*n^(i-1),i=1..nops(L)))
end proc:
select(filter, [seq(i,i=11..2000,2)]); # Robert Israel, Dec 04 2016
  • PARI
    isok(n) = (n>=10) && isprime(fromdigits(digits(n), n)); \\ Michel Marcus, Dec 04 2016

Extensions

Offset corrected by Arkadiusz Wesolowski, Oct 19 2013
Name corrected by Robert Israel, Dec 04 2016

A294601 Numbers with exactly one odd decimal digit.

Original entry on oeis.org

1, 3, 5, 7, 9, 10, 12, 14, 16, 18, 21, 23, 25, 27, 29, 30, 32, 34, 36, 38, 41, 43, 45, 47, 49, 50, 52, 54, 56, 58, 61, 63, 65, 67, 69, 70, 72, 74, 76, 78, 81, 83, 85, 87, 89, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 120, 122, 124, 126, 128, 140, 142, 144, 146, 148, 160, 162, 164, 166, 168, 180
Offset: 1

Views

Author

Robert Israel, Nov 03 2017

Keywords

Comments

First differs from A054684 at position 56.
Numbers n such that A196564(n) = 1. - Felix Fröhlich, Nov 03 2017
There are (1+4*d)*5^(d-1) = 5*A081040(d+1) terms with d digits. - Robert Israel, Nov 06 2017

Links

Crossrefs

Cf. A014263, A054684, A081040, A196564, A275775.

Programs

  • Maple
    Res:= NULL:
for t from 0 to 1000 do
  if nops(select(type,convert(t,base,10),odd))=1 then Res:= Res,t fi
od:
Res;
  • Mathematica
    Select[Range@ 200, Count[IntegerDigits@ #, ?OddQ] == 1 &] (* _Michael De Vlieger, Nov 03 2017 *)
  • PARI
    a196564(n) = #select(x->x%2, digits(n)) \\ after Michel Marcus
is(n) = a196564(n)==1 \\ Felix Fröhlich, Nov 03 2017
Showing 1-10 of 12 results. Next

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