A356929 -id:A356929 - cp's OEIS frontend

A054683 Numbers whose sum of digits is even.

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

Odimar Fabeny, Apr 19 2000

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

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for 10-automatic sequences.
Index entries for Colombian or self numbers and related sequences

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).

Select[Range[0,200],EvenQ[Total[IntegerDigits[#]]]&] (* Harvey P. Dale, Jan 04 2015 *)

is(n)=my(d=digits(n));sum(i=1,#d,d[i])%2==0 \\ Charles R Greathouse IV, Aug 09 2013

a(n) = n--; m = 10*(n\5); s=sumdigits(m); m + (1-(s-1)%2) + 2*(n%5) \\ David A. Corneth, Jun 05 2016

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

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

More terms from James Sellers, Apr 19 2000

Example corrected by David A. Corneth, Jun 05 2016

A054684 Numbers whose sum of digits is odd.

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, 111, 113, 115, 117, 119, 120, 122, 124, 126, 128, 131

Offset: 1

Odimar Fabeny, Apr 19 2000

Union of A179083 and A179085; A179081(a(n)) = 1. - Reinhard Zumkeller, Jun 28 2010

Equivalently, integers with an odd number of odd digits. - Bernard Schott, Nov 06 2022

			1, 3, 5, 7, 9, 10(1), 12(3), 14(5), 16(7), 18(9), 21(3) and so on.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for 10-automatic sequences.
Index entries for Colombian or self numbers and related sequences

Cf. A054683, A137233 (number of n-digits terms).
Cf. A356929 (even number of even digits).
A294601 (exactly one odd decimal digit) is a subsequence.

[seq(`if`(convert(convert(2*n-1,base,10),`+`)::odd, 2*n-1, 2*n-2), n=1..501)];

Select[Range[200],OddQ[Total[IntegerDigits[#]]]&] (* Harvey P. Dale, Nov 27 2021 *)

is(n)=my(d=digits(n));sum(i=1,#d,d[i])%2 \\ Charles R Greathouse IV, Aug 09 2013

isok(m) = sumdigits(m) % 2; \\ Michel Marcus, Nov 06 2022

a(n) = n=2*(n-1); n + !(sumdigits(n)%2); \\ Kevin Ryde, Nov 07 2022

def ok(n): return sum(map(int, str(n)))&1
print([k for k in range(132) if ok(k)]) # Michael S. Branicky, Nov 06 2022

a(n) = n * 2 - 1 for the first 5 numbers; a(n) = n * 2 for the second 5 numbers.
From Robert Israel, Jun 27 2017: (Start)
a(n) = 2*n-2 if floor((n-1)/5) is in the sequence, 2*n-1 if not.
G.f. g(x) satisfies g(x) = (1-x)*(1+x+x^2+x^3+x^4)^2*g(x^10)/x^9 + x^2*(2+x^4+3*x^5-x^9+3*x^10)/((1-x)*(1+x^5))^2.
(End)

More terms from James Sellers, Apr 19 2000

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

Ctibor O. Zizka, Mar 08 2008

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

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

Olympiade Mathématique Belge, OMB 2004, Finale Maxi, Question 2.
Index to sequences related to Olympiads.
Index entries for linear recurrences with constant coefficients, signature (10).

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

Cf. A030656, A052268, A054684, A356929.

Vec(5*x*(1-x)/(1-10*x) + O(x^22)) \\ Elmo R. Oliveira, Jul 23 2025

def A137233(n): return 9*10**(n-1)+1>>1 # Chai Wah Wu, Nov 11 2022

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)

Corrected and extended by R. J. Mathar, May 23 2008

More terms from Elmo R. Oliveira, Jul 23 2025

A358270 Numbers whose sum of digits is even and that have an even number of even digits.

Original entry on oeis.org

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, 1001, 1003, 1005, 1007, 1009, 1010, 1012, 1014, 1016, 1018, 1021, 1023, 1025, 1027, 1029, 1030

Offset: 1

Bernard Schott, Nov 06 2022

There are only terms with an even number of digits, and precisely, there exist A137233(2*k) terms with 2*k digits.
The conditions separately are A054683 for even sum of digits, and A356929 for even number of even digits, so that this sequence is their intersection.
The opposite conditions, an odd sum of digits, and an odd number of odd digits, are the same and are A054684.

			26 is a term since 2+6 = 8 (even) and 26 has two even digits.
39 is a term since 3+9 = 12 (even) and 39 has zero even digits.
1012 is a term since 1+0+1+2 = 4 (even) and 1012 has two even digits.

Intersection of A054683 and A356929.
Cf. A001637 (even length), A179081 (digit sum mod 2).
Cf. A014263, A054684, A137233.

Select[Range[1000], EvenQ[Plus @@ IntegerDigits[#]] && EvenQ[Plus @@ DigitCount[#, 10, Range[0, 8, 2]]] &] (* Amiram Eldar, Nov 06 2022 *)

a(n) = n*=2; n += 100^logint(110*n,100) \ 11; n - sumdigits(n)%2; \\ Kevin Ryde, Nov 10 2022

def ok(n): s = str(n); return sum(map(int, s))%2 == sum(1 for d in s if d in "02468")%2 == 0
print([k for k in range(1031) if ok(k)]) # Michael S. Branicky, Nov 06 2022

from itertools import count, islice, chain
def A358270_gen(): # generator of terms
    return filter(lambda n:not (len(s:=str(n))&1 or sum(int(d) for d in s)&1), chain.from_iterable((range(10**l,10**(l+1)) for l in count(1,2))))
A358270_list = list(islice(A358270_gen(),61)) # Chai Wah Wu, Nov 11 2022

a(n) = t - A179081(t) where t = A001637(2*n). - Kevin Ryde, Nov 10 2022

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A054683 Numbers whose sum of digits is even.

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A054684 Numbers whose sum of digits is odd.

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A137233 Number of n-digit even numbers.

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A358270 Numbers whose sum of digits is even and that have an even number of even digits.

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