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.

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A014263 Numbers that contain even digits only.

Original entry on oeis.org

0, 2, 4, 6, 8, 20, 22, 24, 26, 28, 40, 42, 44, 46, 48, 60, 62, 64, 66, 68, 80, 82, 84, 86, 88, 200, 202, 204, 206, 208, 220, 222, 224, 226, 228, 240, 242, 244, 246, 248, 260, 262, 264, 266, 268, 280, 282, 284, 286, 288, 400, 402, 404, 406, 408, 420, 422, 424
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

The set of real numbers between 0 and 1 that contain no odd digits in their decimal expansion has Hausdorff dimension log 5 / log 10.
Integers written in base 5 and then doubled (in base 10). - Franklin T. Adams-Watters, Mar 15 2006
The carryless mod 10 "even" numbers (cf. A004529) sorted and duplicates removed. - N. J. A. Sloane, Aug 03 2010.
Complement of A007957; A196564(a(n)) = 0; A103181(a(n)) = 0. - Reinhard Zumkeller, Oct 04 2011
If n-1 is represented as a base-5 number (see A007091) 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)=0,2,4,6,8 for k=0..4. - Hieronymus Fischer, Jun 03 2012

Examples

			a(1000) = 24888.
a(10^4) = 60888.
a(10^5) = 22288888.
a(10^6) = 446888888.

References

  • K. J. Falconer, The Geometry of Fractal Sets, Cambridge, 1985; p. 19.

Links

Crossrefs

Subsequence of A059708.
Cf. A061810, A061811, A007091, A014261, A046034, A052382, A084544, A089581, A084984, A017042, A001743, A202267, A202268, A194182, A196563.

Programs

  • Haskell
    a014263 n = a014263_list !! (n-1)
a014263_list = filter (all (`elem` "02468") . show) [0,2..]
-- Reinhard Zumkeller, Jul 05 2011
  • Magma
    [n: n in [0..424] | Set(Intseq(n)) subset [0..8 by 2]];  // Bruno Berselli, Jul 19 2011
  • Maple
    a:= proc(m) local L,i;
  L:= convert(m-1,base,5);
  2*add(L[i]*10^(i-1),i=1..nops(L))
end proc:
seq(a(i),i=1..100); # Robert Israel, Apr 07 2016
  • Mathematica
    Select[Range[450], And@@EvenQ[IntegerDigits[#]]&] (* Harvey P. Dale, Jan 30 2011 *)
FromDigits/@Tuples[{0,2,4,6,8},3] (* Harvey P. Dale, Jul 07 2025 *)
  • PARI
    a(n) = 2*fromdigits(digits(n-1, 5), 10); \\ Michel Marcus, Nov 04 2022
  • PARI
    is(n)=#setminus(Set(digits(n)), [0,2,4,6,8])==0 \\ Charles R Greathouse IV, Mar 03 2025
  • Python
    from sympy.ntheory.digits import digits
def a(n): return int(''.join(str(2*d) for d in digits(n, 5)[1:]))
print([a(n) for n in range(58)]) # Michael S. Branicky, Jan 13 2022
  • Python
    from itertools import count, islice, product
def agen(): # generator of terms
    yield 0
    for d in count(1):
        for first in "2468":
            for rest in product("02468", repeat=d-1):
                yield int(first + "".join(rest))
print(list(islice(agen(), 58))) # Michael S. Branicky, Jan 13 2022

Formula

A045888(a(n)) = 0. - Reinhard Zumkeller, Aug 25 2009
a(n) = A179082(n) for n <= 25. - Reinhard Zumkeller, Jun 28 2010
From Hieronymus Fischer, Jun 06 2012: (Start)
a(n) = ((2*b_m(n)) mod 8 + 2)*10^m + Sum_{j=0..m-1} ((2*b_j(n)) mod 10)*10^j, where n>1, b_j(n) = floor((n-1-5^m)/5^j), m = floor(log_5(n-1)).
a(1*5^n+1) = 2*10^n.
a(2*5^n+1) = 4*10^n.
a(3*5^n+1) = 6*10^n.
a(4*5^n+1) = 8*10^n.
a(n) = 2*10^log_5(n-1) for n=5^k+1,
a(n) < 2*10^log_5(n-1), else.
a(n) > (8/9)*10^log_5(n-1) n>1.
a(n) = 2*A007091(n-1), iff the digits of A007091(n-1) are 0 or 1.
G.f.: g(x) = (x/(1-x))*Sum_{j>=0} 10^j*x^5^j *(1-x^5^j)* (2+4x^5^j+ 6(x^2)^5^j+ 8(x^3)^5^j)/(1-x^5^(j+1)).
Also: g(x) = 2*(x/(1-x))*Sum_{j>=0} 10^j*x^5^j * (1-4x^(3*5^j)+3x^(4*5^j))/((1-x^5^j)(1-x^5^(j+1))).
Also: g(x) = 2*(x/(1-x))*(h_(5,1)(x) + h_(5,2)(x) + h_(5,3)(x) + h_(5,4)(x) - 4*h_(5,5)(x)), where h_(5,k)(x) = Sum_{j>=0} 10^j*(x^5^j)^k/(1-(x^5^j)^5). (End)
a(5*n+i-4) = 10*a(n) + 2*i for n >= 1, i=0..4. - Robert Israel, Apr 07 2016
Sum_{n>=2} 1/a(n) = A194182. - Bernard Schott, Jan 13 2022

Extensions

Examples and crossrefs added by Hieronymus Fischer, Jun 06 2012

A194181 Decimal expansion of the (finite) value of Sum_{k >= 1, k has no even digit in base 10 } 1/k.

Original entry on oeis.org

3, 1, 7, 1, 7, 6, 5, 4, 7, 3, 4, 1, 5, 9, 0, 4, 9, 5, 7, 2, 2, 8, 7, 0, 9, 7, 0, 8, 7, 5, 0, 6, 1, 1, 6, 5, 6, 7, 9, 7, 0, 5, 0, 7, 0, 8, 3, 9, 6, 2, 8, 5, 7, 2, 4, 1, 6, 4, 1, 8, 6, 8, 9, 8, 4, 3, 7, 1, 3, 7, 6, 8, 8, 5, 8, 5, 6, 1, 9, 2, 6, 6, 8, 8, 5, 2, 3, 1, 0, 8, 0, 7, 4, 7, 1, 5, 6, 0, 4, 5, 4
Offset: 1

Views

Author

Robert G. Wilson v, Aug 18 2011

Keywords

Comments

For an elementary proof that this series is convergent, see Honsberger's reference. - Bernard Schott, Jan 13 2022

Examples

			3.17176547341590495722870970875061165679705070839628572416418689843...

References

  • Ross Honsberger, Mathematical Gems II, Dolciani Mathematical Expositions No. 2, Mathematical Association of America, 1976, pp. 102 and 177.

Links

Crossrefs

Cf. A014261, A082830, A194182.

Programs

  • Mathematica
    RealDigits[kSum[{0, 2, 4, 6, 8}, 120 ]][[1]] (* Amiram Eldar, Jun 15 2023, using Baillie and Schmelzer's kempnerSums.nb, see Links *)

Formula

Equals Sum_{n>=1} 1/A014261(n). - Bernard Schott, Jan 13 2022

A258271 Decimal expansion of the sum of the reciprocal of the squares of the numbers whose digits are all even.

Original entry on oeis.org

0, 3, 6, 6, 3, 6, 0, 0, 3, 9, 7, 1, 9, 5, 2, 3, 2, 9, 5, 1, 7, 1, 8, 8, 2, 5, 0, 8, 9, 6, 7, 4, 1, 2, 4, 2, 6, 6, 2, 5, 1, 7, 3, 9, 5, 0, 3, 4, 2, 1, 1, 8, 7, 6, 0, 0, 2, 0, 0, 7, 1, 1, 3, 5, 0, 8, 5, 2, 8, 3, 3, 3, 2, 9, 3, 4, 9, 5, 1, 5, 7, 5, 8, 4, 4, 6, 5
Offset: 1

Views

Author

Paolo P. Lava, May 25 2015

Keywords

Comments

A rational approximation (correct up to the 9th decimal digit) is 22781/62182.
Continued fraction: [0, 2, 1, 2, 1, 2, 3, 3, 1, 8, 5, 2, 1, 14,...].

Examples

			Decimal expansion of Sum_{k=1..oo}{1/A045926(k)^2} = 1/2^2 + 1/4^2 + 1/6^2 + 1/8^2 + 1/22^2 + 1/24^2 + 1/26^2 + ... = 0.3663600397195232951718825089674124266251739503421187600...

Crossrefs

Cf. A045926, A194181, A194182.

Programs

  • Maple
    P:=proc(q) local a,b,k,ok,n; a:=0; for n from 2 by 2 to q do ok:=1; b:=n;
for k from 1 to ilog10(n)+1 do if (b mod 10)=0 or ((b mod 10) mod 2)=1 then ok:=0;
break; else b:=trunc(b/10); fi; od; if ok=1 then a:=a+(1/n)^2; fi; od;
print(evalf(a,200)); end: P(10^9);
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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