A014263 -id:A014263 - 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

A202268 Numbers in which all digits are nonprimes (1, 4, 6, 8, 9).

Original entry on oeis.org

1, 4, 6, 8, 9, 11, 14, 16, 18, 19, 41, 44, 46, 48, 49, 61, 64, 66, 68, 69, 81, 84, 86, 88, 89, 91, 94, 96, 98, 99, 111, 114, 116, 118, 119, 141, 144, 146, 148, 149, 161, 164, 166, 168, 169, 181, 184, 186, 188, 189, 191, 194, 196, 198, 199, 411, 414, 416, 418, 419

Offset: 1

Jaroslav Krizek, Dec 25 2011

Supersequence of A029581.
Subsequence of A084984.
If n-1 is represented as a zerofree base-5 number (see A084545) according to n-1=d(m)d(m-1)...d(3)d(2)d(1)d(0) then a(n) = Sum_{j=0..m} c(d(j))*10^j, where c(k)=1,4,6,8,9 for k=1..5. - Hieronymus Fischer, May 30 2012

			From _Hieronymus Fischer_, May 30 2012: (Start)
a(1000) = 14889.
a(10^4) = 498889
a(10^5) = 11188889.
a(10^6) = 446888889. (End)

Hieronymus Fischer, Table of n, a(n) for n = 1..10000
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Index entries for 10-automatic sequences.

Cf. A046034 (numbers in which all digits are primes), A001742 (numbers in which all digits are noncomposites excluding 0), A202267 (numbers in which all digits are noncomposites), A084984 (numbers in which all digits are nonprimes), A029581 (numbers in which all digits are composites).

Cf. A084545, A001743, A001744, A014261, A014263.

[n: n in [1..500] | Set(Intseq(n)) subset [1, 4, 6, 8, 9]]; // Vincenzo Librandi, Dec 17 2018

Table[FromDigits/@Tuples[{1, 4, 6, 8, 9}, n], {n, 3}] // Flatten (* Vincenzo Librandi, Dec 17 2018 *)

From Hieronymus Fischer, May 30 2012: (Start)
a(n) = Sum_{j=0..m-1} ((2*b_j(n)+1) mod 10 + floor((b_j(n)+4)/5) - floor((b_j(n)+1)/5))*10^j, where b_j(n))=floor((4*n+1-5^m)/(4*5^j)), m=floor(log_5(4*n+1)).
a(1*(5^n-1)/4) = 1*(10^n-1)/9.
a(2*(5^n-1)/4) = 4*(10^n-1)/9.
a(3*(5^n-1)/4) = 6*(10^n-1)/9.
a(4*(5^n-1)/4) = 8*(10^n-1)/9.
a(5*(5^n-1)/4) = 10^n-1.
a(n) = (10^log_5(4*n+1)-1)/9 for n=(5^k-1)/4, k>0.
a(n) <= 36/(9*2^log_5(9)-1)*(10^log_5(4*n+1)-1)/9 for n>0, equality holds for n=2.
a(n) > 0.776*10^log_5(4*n+1)-1)/9 for n>0.
a(n) >= A001742(n), equality holds for n=(5^k-1)/4, k>0.
a(n) = A084545(n) iff all digits of A084545(n) are 1, a(n)>A084545(n), else.
G.f.: g(x) = (x^(1/4)*(1-x))^(-1) Sum_{j>=0} 10^j*z(j)^(5/4)*(1-z(j))*(1 + 4z(j) + 6*z(j)^2 + 8*z(j)^3 + 9*z(j)^4)/(1-z(j)^5), where z(j)=x^5^j.
Also: g(x) = (1/(1-x))*(h_(5,0)(x) + 3h_(5,1)(x) + 2h_(5,2)(x) + 2h_(5,3)(x) + h_(5,4)(x) - 9*h_(5,5)(x)), where h_(5,k)(x) = Sum_{j>=0} 10^j*x^((5^(j+1)-1)/4)*(x^5^j)^k/(1-(x^5^j)^5). (End)
Sum_{n>=1} 1/a(n) = 2.897648425695540438556738520657902585305276107220152307051361916356295164643... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 15 2024

A169964 Numbers whose decimal expansion contains only 0's and 5's.

Original entry on oeis.org

0, 5, 50, 55, 500, 505, 550, 555, 5000, 5005, 5050, 5055, 5500, 5505, 5550, 5555, 50000, 50005, 50050, 50055, 50500, 50505, 50550, 50555, 55000, 55005, 55050, 55055, 55500, 55505, 55550, 55555, 500000, 500005, 500050, 500055, 500500, 500505, 500550, 500555

Offset: 1

N. J. A. Sloane, Aug 07 2010

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to carryless arithmetic

Cf. A007088, A014263, A169965, A169966, A169967.

Cf. A078244, A204093, A204094, A204095, A097256.

a169964 n = a169964_list !! (n-1)
a169964_list = map (* 5) a007088_list
-- Reinhard Zumkeller, Jan 10 2012

Map[FromDigits,Tuples[{0,5},6]] (* Paolo Xausa, Oct 30 2023 *)

print1(0);for(d=1,5,for(n=2^(d-1),2^d-1,print1(", ");forstep(i=d-1,0,-1,print1((n>>i)%2*5)))) \\ Charles R Greathouse IV, Nov 16 2011

a(n+1) = Sum_{k>=0} A030308(n,k)*A093143(k+1). - Philippe Deléham, Oct 16 2011

a(n) = 5 * A007088(n-1).

A001742 Numbers whose digits contain no loops (version 2).

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 12, 13, 15, 17, 21, 22, 23, 25, 27, 31, 32, 33, 35, 37, 51, 52, 53, 55, 57, 71, 72, 73, 75, 77, 111, 112, 113, 115, 117, 121, 122, 123, 125, 127, 131, 132, 133, 135, 137, 151, 152, 153, 155, 157, 171, 172, 173, 175, 177, 211, 212, 213, 215

Offset: 1

N. J. A. Sloane

Numbers all of whose decimal digits are in {1,2,3,5,7}.

If n is represented as a zerofree base-5 number (see A084545) according to n = d(m)d(m-1)...d(3)d(2)d(1)d(0) then a(n) = Sum_{j=0..m} c(d(j))*10^j, where c(k)=1,2,3,5,7 for k=1..5. - Hieronymus Fischer, May 30 2012

			From _Hieronymus Fischer_, May 30 2012: (Start)
a(10^3) = 12557.
a(10^4) = 275557.
a(10^5) = 11155557.
a(10^6) = 223555557. (End)

Hieronymus Fischer, Table of n, a(n) for n = 1..10000
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Index entries for 10-automatic sequences.

Cf. A001729 (version 1), A190222 (noncomposite terms), A190223 (n with all divisors in this sequence).

Cf. A046034, A084545, A029581, A084984, A001743, A001744, A014261, A014263, A202267, A202268.

[n: n in [1..500] |  Set(Intseq(n)) subset [1, 2, 3, 5, 7]]; // Vincenzo Librandi, Dec 17 2018

nlQ[n_]:=And@@(MemberQ[{1,2,3,5,7},#]&/@IntegerDigits[n]); Select[Range[ 160],nlQ] (* Harvey P. Dale, Mar 23 2012 *)
Table[FromDigits/@Tuples[{1, 2, 3, 5, 7}, n], {n, 3}] // Flatten (* Vincenzo Librandi, Dec 17 2018 *)

for (my $k = 1; $k < 1000; $k++) {print "$k, " if ($k =~ m/^[12357]+$/)} # Charles R Greathouse IV, Jun 10 2011

From Hieronymus Fischer, May 30 2012: (Start)
a(n) = Sum_{j=0..m-1} ((2*b_j(n)+1) mod 10 + 2*floor(b_j(n)/5) - floor((b_j(n)+3)/5) - floor((b_j(n)+4)/5))*10^j, where b_j(n) = floor((4*n+1-5^m)/(4*5^j)), m = floor(log_5(4*n+1)).
a(1*(5^n-1)/4) = 1*(10^n-1)/9.
a(2*(5^n-1)/4) = 2*(10^n-1)/9.
a(3*(5^n-1)/4) = 1*(10^n-1)/3.
a(4*(5^n-1)/4) = 5*(10^n-1)/9.
a(5*(5^n-1)/4) = 7*(10^n-1)/9.
a(n) = (10^log_5(4*n+1)-1)/9 for n=(5^k-1)/4, k > 0.
a(n) < (10^log_5(4*n+1)-1)/9 for (5^k-1)/4 < n < (5^(k+1)-1)/4, k > 0.
a(n) <= A202268(n), equality holds for n=(5^k-1)/4, k > 0.
a(n) = A084545(n) iff all digits of A084545(n) are <= 3, a(n) > A084545(n), otherwise.
G.f.: g(x) = (x^(1/4)*(1-x))^(-1) Sum_{j>=0} 10^j*z(j)^(5/4)*(1 + z(j) + z(j)^2 + 2*z(j)^3 + 2*z(j)^4 - 7*z(j)^5)/(1-z(j)^5), where z(j) = x^5^j.
Also g(x) = (x^(1/4)*(1-x))^(-1) Sum_{j>=0} 10^j*z(j)^(5/4)*(1-z(j))*(1 + 2z(j) + 3*z(j)^2 + 5*z(j)^3 + 7*z(j)^4)/(1-z(j)^5), where z(j) = x^5^j.
Also: g(x)=(1/(1-x))*(h_(5,0)(x) + h_(5,1)(x) + h_(5,2)(x) + 2*h_(5,3)(x) + 2*h_(5,4)(x) - 7*h_(5,5)(x)), where h_(5,k)(x) = Sum_{j>=0} 10^j*x^((5^(j+1)-1)/4)*(x^5^j)^k/(1-(x^5^j)^5). (End)
Sum_{n>=1} 1/a(n) = 3.961674246441345455010500439753914974057344229353697593567607096540565407371... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 15 2024

A059708 Numbers k such that all digits have same parity.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 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, 111, 113, 115, 117, 119, 131, 133, 135, 137, 139

Offset: 1

N. J. A. Sloane, Feb 07 2001

A059717(a(n)) = a(n). - Reinhard Zumkeller, Jul 05 2011

A059707(a(n)) = a(n). - Reinhard Zumkeller, Jun 15 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Union of A014261 and A014263.

Cf. A059707, A059717, A061383.

a059708 n = a059708_list !! (n-1)
a059708_list = filter sameParity [0..] where
   sameParity n = all (`elem` "02468") ns
               || all (`elem` "13579") ns where ns = show n
-- Reinhard Zumkeller, Jul 05 2011

Select[Range[0, 140], (cnt = Count[id = IntegerDigits[#], ?OddQ]; cnt == 0 || cnt == Length[id]) & ] (* _Jean-François Alcover, May 16 2013 *)

A007932 Numbers that contain only 1's, 2's and 3's.

Original entry on oeis.org

1, 2, 3, 11, 12, 13, 21, 22, 23, 31, 32, 33, 111, 112, 113, 121, 122, 123, 131, 132, 133, 211, 212, 213, 221, 222, 223, 231, 232, 233, 311, 312, 313, 321, 322, 323, 331, 332, 333, 1111, 1112, 1113, 1121, 1122, 1123, 1131, 1132, 1133, 1211, 1212, 1213, 1221

Offset: 1

R. Muller

This sequence is the alternate number system in base 3. - Robert R. Forslund (forslund(AT)tbaytel.net), Jun 27 2003
a(n) is the "bijective base-k numeration" or "k-adic notation" for k=3. - Chris Gaconnet (gaconnet(AT)gmail.com), May 27 2009
a(n) = n written in base 3 where zeros are not allowed but threes are. The three distinct digits used are 1, 2 and 3 instead of 0, 1 and 2. To obtain this sequence from the "canonical" base 3 sequence with zeros allowed, just replace any 0 with a 3 and then subtract one from the group of digits situated on the left: (20-->13; 100-->23; 110-->33; 1000-->223; 1010-->233). This can be done in any integer positive base b, replacing zeros with positive b's and subtracting one from the group of digits situated on the left. And zero is the only digit that can be replaced, since there is always a more significant digit greater than 0, on the left, from which to subtract one. - Robin Garcia, Jan 07 2014

			a(100)  = 3131.
a(10^3) = 323231.
a(10^4) = 111123331.
a(10^5) = 11231311131.
a(10^6) = 1212133131231.
a(10^7) = 123133223331331.
a(10^8) = 13221311111312131.
a(10^9) = 2113123122313232231.
- _Hieronymus Fischer_, Jun 06 2012

K. Atanassov, On the 97th, 98th and the 99th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 3, 89-93.
A. Salomaa, Formal Languages, Academic Press, 1973. pages 90-91. [From Chris Gaconnet (gaconnet(AT)gmail.com), May 27 2009]

Hieronymus Fischer, Table of n, a(n) for n = 1..10000
K. Atanassov, On Some of Smarandache's Problems
EMIS, Mirror site for Southwest Journal of Pure and Applied Mathematics
Robert R. Forslund, A Logical Alternative to the Existing Positional Number System, Southwest Journal of Pure and Applied Mathematics. Vol. 1 1995 pp. 27-29.
James E. Foster, A Number System without a Zero-Symbol, Mathematics Magazine, Vol. 21, No. 1. (1947), pp. 39-41.
F. Smarandache, Only Problems, Not Solutions!.
Wikipedia, Bijective numeration
Index entries for 10-automatic sequences.

Cf. A007931, A052382, A084544, A084545, A046034, A089581, A084984, A001742, A001743, A001744, A202267, A202268, A014261, A014263.

NextNbr[n_] := Block[{d = IntegerDigits[n + 1], l}, l = Length[d]; While[l != 1, If[ d[[l]] > 3, d[[l - 1]]++; d[[l]] = 1]; l-- ]; If[ d[[1]] > 3, d[[1]] = 11]; FromDigits[d]]; NestList[ NextNbr, 1, 51]
Table[FromDigits/@Tuples[{1,2,3},n],{n,4}]//Flatten (* Harvey P. Dale, Mar 29 2018 *)

a(n) = my (w=3); while (n>w, n -= w; w *= 3); my (d=digits(w+n-1, 3)); d[1] = 0; fromdigits(d) + (10^(#d-1)-1)/9 \\ Rémy Sigrist, Aug 28 2018

From Hieronymus Fischer, May 30 2012 and Jun 08 2012: (Start)
The formulas are designed to calculate base-10 numbers only using the digits 1, 2, 3.
a(n) = Sum_{j=0..m-1} (1 + b(j) mod 3)*10^j,
where m = floor(log_3(2*n+1)), b(j) = floor((2*n+1-3^m)/(2*3^j)).
Special values:
a(k*(3^n-1)/2) = k*(10^n-1)/9, k=1,2,3.
a((5*3^n-3)/2) = (4*10^n-1)/3 = 10^n + (10^n-1)/3.
a((3^n-1)/2 - 1) = (10^(n-1)-1)/3, n>1.
Inequalities:
a(n) <= (10^log_3(2*n+1)-1)/9, equality holds for n=(3^k-1)/2, k>0.
a(n) > (3/10)*(10^log_3(2*n+1)-1)/9, n>0.
Lower and upper limits:
lim inf a(n)/10^log_3(2*n) = 1/30, for n --> infinity.
lim sup a(n)/10^log_3(2*n) = 1/9, for n --> infinity.
G.f.: g(x) = (x^(1/2)*(1-x))^(-1) Sum_{j=>0} 10^j*(x^3^j)^(3/2) * (1-x^3^j)*(1 + 2x^3^j + 3x^(2*3^j))/(1 - x^3^(j+1)).
Also: g(x) = (1/(1-x)) Sum_{j>=0} (1 - 4(x^3^j)^3 + 3(x^3^j)^4)*x^3^j*f_j(x)/(1-x^3^j), where f_j(x) = 10^j*x^((3^j-1)/2)/(1-(x^3^j)^3). The f_j obey the recurrence f_0(x) = 1/(1-x^3), f_(j+1)(x) = 10x*f_j(x^3).
Also: g(x) = (1/(1-x))*(h_(3,0)(x) + h_(3,1)(x) + h_(3,2)(x) - 3*h_(3,3)(x)), where h_(3,k)(x) = Sum_{j>=0} 10^j*x^((3^(j+1)-1)/2) * (x^3^j)^k/(1-(x^3^j)^3).
(End)

Edited and extended by Robert G. Wilson v, Dec 14 2002

Crossrefs added by Hieronymus Fischer, Jun 06 2012

A169965 Numbers whose decimal expansion contains only 0's and 2's.

Original entry on oeis.org

0, 2, 20, 22, 200, 202, 220, 222, 2000, 2002, 2020, 2022, 2200, 2202, 2220, 2222, 20000, 20002, 20020, 20022, 20200, 20202, 20220, 20222, 22000, 22002, 22020, 22022, 22200, 22202, 22220, 22222, 200000, 200002, 200020, 200022, 200200, 200202, 200220, 200222

Offset: 1

N. J. A. Sloane, Aug 07 2010

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to carryless arithmetic

Cf. A007088, A014263, A169964, A169966, A169967.

Cf. A078241, A204093, A204094, A204095, A097256.

a169965 n = a169965_list !! (n-1)
a169965_list = map (* 2) a007088_list
-- Reinhard Zumkeller, Jan 10 2012

Map[FromDigits,Tuples[{0,2},6]] (* Paolo Xausa, Oct 30 2023 *)

print1(0);for(d=1,5,for(n=2^(d-1),2^d-1,print1(", ");forstep(i=d-1,0,-1,print1((n>>i)%2*2)))) \\ Charles R Greathouse IV, Nov 16 2011

lista(N) = vector(N, i, fromdigits(binary(i-1)*2)); \\ Ruud H.G. van Tol, Oct 26 2024

a(n+1) = Sum_{k>=0} A030308(n,k)*A093136(k+1). - Philippe Deléham, Oct 16 2011

a(n) = 2 * A007088(n-1).

A169967 Numbers whose decimal expansion contains only 0's and 4's.

Original entry on oeis.org

0, 4, 40, 44, 400, 404, 440, 444, 4000, 4004, 4040, 4044, 4400, 4404, 4440, 4444, 40000, 40004, 40040, 40044, 40400, 40404, 40440, 40444, 44000, 44004, 44040, 44044, 44400, 44404, 44440, 44444, 400000, 400004, 400040, 400044, 400400, 400404, 400440, 400444

Offset: 1

N. J. A. Sloane, Aug 07 2010

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Cf. A007088, A014263, A093141, A169964, A169965, A169966, A078243, A204093, A204094, A204095, A097256.

a169967 n = a169967_list !! (n-1)
a169967_list = map (* 4) a007088_list
-- Reinhard Zumkeller, Jan 10 2012

FromDigits/@Tuples[{0,4},6] (* Harvey P. Dale, Dec 21 2018 *)

print1(0);for(d=1,5,for(n=2^(d-1),2^d-1,print1(", ");forstep(i=d-1,0,-1,print1((n>>i)%2*4)))) \\ Charles R Greathouse IV, Nov 16 2011

a(n+1) = Sum_{k>=0} A030308(n,k)*A093141(k+1). - Philippe Deléham, Oct 16 2011

a(n) = 4 * A007088(n-1).

A084544 Alternate number system in base 4.

Original entry on oeis.org

1, 2, 3, 4, 11, 12, 13, 14, 21, 22, 23, 24, 31, 32, 33, 34, 41, 42, 43, 44, 111, 112, 113, 114, 121, 122, 123, 124, 131, 132, 133, 134, 141, 142, 143, 144, 211, 212, 213, 214, 221, 222, 223, 224, 231, 232, 233, 234, 241, 242, 243, 244, 311, 312, 313, 314, 321

Offset: 1

Robert R. Forslund (forslund(AT)tbaytel.net), Jun 27 2003

			From _Hieronymus Fischer_, Jun 06 2012: (Start)
a(100)  = 1144.
a(10^3) = 33214.
a(10^4) = 2123434.
a(10^5) = 114122134.
a(10^6) = 3243414334.
a(10^7) = 211421121334.
a(10^8) = 11331131343334.
a(10^9) = 323212224213334. (End)

Hieronymus Fischer, Table of n, a(n) for n = 1..10000
EMIS, Mirror site for Southwest Journal of Pure and Applied Mathematics
R. R. Forslund, A logical alternative to the existing positional number system, Southwest Journal of Pure and Applied Mathematics, Vol. 1 1995 pp. 27-29.
R. R. Forslund, Positive Integer Pages [Broken link]
James E. Foster, A Number System without a Zero-Symbol, Mathematics Magazine, Vol. 21, No. 1. (1947), pp. 39-41.
Index entries for 10-automatic sequences.

Cf. A007931, A007932, A052382, A084545, A046034, A089581, A084984, A001742, A001743, A001744, A202267, A202268, A014261, A014263.

def A084544(n):
    m = (3*n+1).bit_length()-1>>1
    return int(''.join((str(((3*n+1-(1<<(m<<1)))//(3<<((m-1-j)<<1))&3)+1) for j in range(m)))) # Chai Wah Wu, Feb 08 2023

From Hieronymus Fischer, Jun 06 and Jun 08 2012: (Start)
The formulas are designed to calculate base-10 numbers only using the digits 1..4.
a(n) = Sum_{j=0..m-1} (1 + b(j) mod 4)*10^j,
where m = floor(log_4(3*n+1)), b(j) = floor((3*n+1-4^m)/(3*4^j)).
Special values:
a(k*(4^n-1)/3) = k*(10^n-1)/9, k = 1,2,3,4.
a((7*4^n-4)/3) = (13*10^n-4)/9 = 10^n + 4*(10^n-1)/9.
a((4^n-1)/3 - 1) = 4*(10^(n-1)-1)/9, n > 1.
Inequalities:
a(n) <= (10^log_4(3*n+1)-1)/9, equality holds for n=(4^k-1)/3, k>0.
a(n) > (4/10)*(10^log_4(3*n+1)-1)/9, n > 0.
Lower and upper limits:
lim inf a(n)/10^log_4(3*n) = 2/45, for n --> infinity.
lim sup a(n)/10^log_4(3*n) = 1/9, for n --> infinity.
G.f.: g(x) = (x^(1/3)*(1-x))^(-1) Sum_{j>=0} 10^j*z(j)^(4/3)*(1 - 5z(j)^4 + 4z(j)^5)/((1-z(j))(1-z(j)^4)), where z(j) = x^4^j.
Also: g(x) = (1/(1-x)) Sum_{j>=0} (1-5(x^4^j)^4 + 4(x^4^j)^5)*x^4^j*f_j(x)/(1-x^4^j), where f_j(x) = 10^j*x^((4^j-1)/3)/(1-(x^4^j)^4). The f_j obey the recurrence f_0(x) = 1/(1-x^4), f_(j+1)(x) = 10x*f_j(x^4).
Also: g(x) = (1/(1-x))* (h_(4,0)(x) + h_(4,1)(x) + h_(4,2)(x) + h_(4,3)(x) - 4*h_(4,4)(x)), where h_(4,k)(x) = Sum_{j>=0} 10^j*x^((4^(j+1)-1)/3) * (x^4^j)^k/(1-(x^4^j)^4).
(End)
a(n) = A045926(n) / 2. - Reinhard Zumkeller, Jan 01 2013

Offset set to 1 according to A007931, A007932 by Hieronymus Fischer, Jun 06 2012

A071649 Sum of odd decimal digits of n.

Original entry on oeis.org

1, 0, 3, 0, 5, 0, 7, 0, 9, 1, 2, 1, 4, 1, 6, 1, 8, 1, 10, 0, 1, 0, 3, 0, 5, 0, 7, 0, 9, 3, 4, 3, 6, 3, 8, 3, 10, 3, 12, 0, 1, 0, 3, 0, 5, 0, 7, 0, 9, 5, 6, 5, 8, 5, 10, 5, 12, 5, 14, 0, 1, 0, 3, 0, 5, 0, 7, 0, 9, 7, 8, 7, 10, 7, 12, 7, 14, 7, 16, 0, 1, 0, 3, 0, 5, 0, 7, 0, 9, 9, 10, 9

Offset: 1

Reinhard Zumkeller, May 28 2002

a(n) = A007953(n) - A071648(n).

Cf. A014263, A071650.

A071649 := proc(n)
      local a,d;
      a := 0 ;
      for d in convert(n,base,10) do
        if type(d,'odd') then
            a := a+d ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Feb 02 2015

Table[Total[Select[IntegerDigits[n], OddQ[#] &]],{n,92}] (* Jayanta Basu, May 23 2013 *)

a(n)=my(d=digits(n)); sum(i=1,#d,if(d[i]%2,d[i])) \\ Charles R Greathouse IV, Apr 04 2014

A071649(n)=vecsum(select(d->bittest(d,0), digits(n))) \\ Nearly twice as fast. - M. F. Hasler, Dec 09 2018

A071649 = lambda x: sum(d for d in map(int, str(x)) if d&1) # M. F. Hasler, Dec 07 2022

a(n) = 0 iff n is in A014263. - Bernard Schott, Mar 17 2023

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