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 15 results. Next

A014261 Numbers that contain odd digits only.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 31, 33, 35, 37, 39, 51, 53, 55, 57, 59, 71, 73, 75, 77, 79, 91, 93, 95, 97, 99, 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, 311, 313, 315, 317, 319
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Or, numbers whose product of digits is odd.
Complement of A007928; A196563(a(n)) = 0. - Reinhard Zumkeller, Oct 04 2011
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, 3, 5, 7, 9 for k = 1..5. - Hieronymus Fischer, Jun 06 2012

Examples

			a(10^3) = 13779.
a(10^4) = 397779.
a(10^5) = 11177779.
a(10^6) = 335777779.

Links

Crossrefs

Subsequence of A059708 and of A225985. A066640 and A030096 are subsequences.
Cf. A005408, A010879, A014263, A046034, A084545, A089581, A084984, A001743, A001744, A202267, A202268, A196564.
Cf. A192107, A194181.

Programs

  • Haskell
    a014261 n = a014261_list !! (n-1)
a014261_list = filter (all (`elem` "13579") . show) [1,3..]
-- Reinhard Zumkeller, Jul 05 2011
  • Magma
    [ n : n in [1..129] | IsOdd(&*Intseq(n,10)) ];
  • Mathematica
    Select[Range[400], OddQ[Times@@IntegerDigits[#]] &] (* Alonso del Arte, Feb 21 2014 *)
  • PARI
    is(n)=Set(digits(n)%2)==[1] \\ Charles R Greathouse IV, Jul 06 2017
  • PARI
    a(n)={my(k=1); while(n>5^k, n-=5^k; k++); fromdigits([2*d+1 | d<-digits(5^k+n-1, 5)]) - 3*10^k} \\ Andrew Howroyd, Jan 17 2020
  • Python
    from itertools import islice, count
def A014261(): return filter(lambda n: set(str(n)) <= {'1','3','5','7','9'}, count(1,2))
A014261_list = list(islice(A014261(),20)) # Chai Wah Wu, Nov 22 2021
  • Python
    from itertools import count, islice, product
def agen(): yield from (int("".join(p)) for d in count(1) for p in product("13579", repeat=d))
print(list(islice(agen(), 60))) # Michael S. Branicky, Jan 13 2022

Formula

A121759(a(n)) = a(n); A000035(A007959(a(n))) = 1. - Reinhard Zumkeller, Nov 30 2007
From Reinhard Zumkeller, Aug 30 2009: (Start)
a(n+1) - a(n) = A164898(n). - Reinhard Zumkeller, Aug 30 2009
a(n+1) = h(a(n)) with h(x) = 1 + (if x mod 10 < 9 then x + x mod 2 else 10*h(floor(x/10)));
a(n) = f(n, 1) where f(n, x) = if n = 1 then x else f(n-1, h(x)). (End)
From Hieronymus Fischer, Jun 06 2012: (Start)
a(n) = Sum_{j = 0..m-1} ((2*b_j(n)+1) mod 10)*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) = 1*(10^n-1)/3.
a(3*(5^n-1)/4) = 5*(10^n-1)/9.
a(4*(5^n-1)/4) = 7*(10^n-1)/9.
a(5*(5^n-1)/4) = 10^n - 1.
a((5^n-1)/4 + 5^(n-1)-1) = (10^n-5)/5.
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) <= 27/(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) if and only if all digits of A084545(n) are 1, else a(n) > A084545(n).
G.f.: g(x)= (x^(1/4)*(1-x))^(-1) Sum_{j >= 0} 10^j*z(j)^(5/4)*(1-z(j))*(1 + 3*z(j) + 5*z(j)^2 + 7*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) + 2*h_(5,1)(x) + 2*h_(5,2)(x) + 2*h_(5,3)(x) + 2*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)
a(n) = A225985(A226091(n)). - Reinhard Zumkeller, May 26 2013
Sum_{n>=1} 1/a(n) = A194181. - Bernard Schott, Jan 13 2022

Extensions

More terms from Robert G. Wilson v, Oct 18 2002
Examples and crossrefs added by Hieronymus Fischer, Jun 06 2012

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

A139281 If all digits are the same mod 3, stop; otherwise write down the number formed by the 1 mod 3 digits and the number formed by the 2 mod 3 digits and the number formed by the 3 mod 3 digits and multiply them; repeat.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 11, 2, 3, 14, 5, 6, 17, 8, 9, 0, 2, 22, 6, 8, 25, 2, 14, 28, 8, 30, 3, 6, 33, 2, 5, 36, 2, 8, 39, 0, 41, 8, 2, 44, 0, 8, 47, 6, 36, 0, 5, 52, 5, 0, 55, 30, 5, 58, 0, 60, 6, 2, 63, 8, 30, 66, 8, 6, 69, 0, 71, 14, 2, 74, 5, 8, 77, 30, 63, 0, 8, 82, 8, 6, 85, 6, 30
Offset: 0

Views

Author

Jonathan Vos Post, Jun 06 2008

Keywords

Comments

Modulo 3 analog of A059707. The 1 mod 3 digits = {1,4,7}, 2 mod 3 digits = {2,5,8}, 3 mod 3 digits = {0, 3, 6, 9}. The fixed points begin: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 17, 22, 25, 28, 30, 33, 36, 39, 41, 44, 47, 52, 55, 58.

Examples

			a(57) = 5 because 5 and 7 are different mod 3, so 5*7 = 35; 3 and 5 are different mod 3, so 3*5 = 15; 1 and 5 are different mod 3, so 1*5 = 5, which is a fixed point.

Crossrefs

Cf. A010872, A059707, A059708, A059717.

Extensions

a(52) corrected and sequence extended by Sean A. Irvine, Sep 03 2009

A385292 Numbers whose digits all belong to the same residue class mod 3.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 17, 22, 25, 28, 30, 33, 36, 39, 41, 44, 47, 52, 55, 58, 60, 63, 66, 69, 71, 74, 77, 82, 85, 88, 90, 93, 96, 99, 111, 114, 117, 141, 144, 147, 171, 174, 177, 222, 225, 228, 252, 255, 258, 282, 285, 288, 300, 303, 306, 309, 330, 333, 336, 339, 360, 363, 366
Offset: 1

Views

Author

Stefano Spezia, Jun 24 2025

Keywords

Links

Crossrefs

Similar sequences for other values of the modulo k: A059708 (k=2), this sequence (k=3), A385293 (k=4), A385294 (k=5), A385295 (k=6), A385296 (k=7), A385297 (k=8), A385298 (k=9).

Programs

  • Mathematica
    Select[Range[0,366],Length[DeleteDuplicates[Mod[IntegerDigits[#],3]]] == 1 &]

A385293 Numbers whose digits all belong to the same residue class mod 4.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 15, 19, 22, 26, 33, 37, 40, 44, 48, 51, 55, 59, 62, 66, 73, 77, 80, 84, 88, 91, 95, 99, 111, 115, 119, 151, 155, 159, 191, 195, 199, 222, 226, 262, 266, 333, 337, 373, 377, 400, 404, 408, 440, 444, 448, 480, 484, 488, 511, 515, 519, 551, 555, 559, 591, 595, 599
Offset: 1

Views

Author

Stefano Spezia, Jun 24 2025

Keywords

Links

Crossrefs

Similar sequences for other values of the modulo k: A059708 (k=2), A385292 (k=3), this sequence (k=4), A385294 (k=5), A385295 (k=6), A385296 (k=7), A385297 (k=8), A385298 (k=9).

Programs

  • Mathematica
    Select[Range[0,600],Length[DeleteDuplicates[Mod[IntegerDigits[#],4]]] == 1 &]

A385294 Numbers whose digits all belong to the same residue class mod 5.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 16, 22, 27, 33, 38, 44, 49, 50, 55, 61, 66, 72, 77, 83, 88, 94, 99, 111, 116, 161, 166, 222, 227, 272, 277, 333, 338, 383, 388, 444, 449, 494, 499, 500, 505, 550, 555, 611, 616, 661, 666, 722, 727, 772, 777, 833, 838, 883, 888, 944, 949, 994, 999, 1111, 1116
Offset: 1

Views

Author

Stefano Spezia, Jun 24 2025

Keywords

Links

Crossrefs

Similar sequences for other values of the modulo k: A059708 (k=2), A385292 (k=3), A385293 (k=4), this sequence (k=5), A385295 (k=6), A385296 (k=7), A385297 (k=8), A385298 (k=9).

Programs

  • Mathematica
    Select[Range[0,1200],Length[DeleteDuplicates[Mod[IntegerDigits[#],5]]] == 1 &]

A385295 Numbers whose digits all belong to the same residue class mod 6.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 17, 22, 28, 33, 39, 44, 55, 60, 66, 71, 77, 82, 88, 93, 99, 111, 117, 171, 177, 222, 228, 282, 288, 333, 339, 393, 399, 444, 555, 600, 606, 660, 666, 711, 717, 771, 777, 822, 828, 882, 888, 933, 939, 993, 999, 1111, 1117, 1171, 1177, 1711, 1717, 1771, 1777, 2222
Offset: 1

Views

Author

Stefano Spezia, Jun 24 2025

Keywords

Links

Crossrefs

Similar sequences for other values of the modulo k: A059708 (k=2), A385292 (k=3), A385293 (k=4), A385294 (k=5), this sequence (k=6), A385296 (k=7), A385297 (k=8), A385298 (k=9).

Programs

  • Mathematica
    Select[Range[0,2300],Length[DeleteDuplicates[Mod[IntegerDigits[#],6]]] == 1 &]

A385296 Numbers whose digits all belong to the same residue class mod 7.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 18, 22, 29, 33, 44, 55, 66, 70, 77, 81, 88, 92, 99, 111, 118, 181, 188, 222, 229, 292, 299, 333, 444, 555, 666, 700, 707, 770, 777, 811, 818, 881, 888, 922, 929, 992, 999, 1111, 1118, 1181, 1188, 1811, 1818, 1881, 1888, 2222, 2229, 2292, 2299, 2922, 2929, 2992, 2999
Offset: 1

Views

Author

Stefano Spezia, Jun 24 2025

Keywords

Links

Crossrefs

Similar sequences for other values of the modulo k: A059708 (k=2), A385292 (k=3), A385293 (k=4), A385294 (k=5), A385295 (k=6), this sequence (k=7), A385297 (k=8), A385298 (k=9).

Programs

  • Mathematica
    Select[Range[0,3000],Length[DeleteDuplicates[Mod[IntegerDigits[#],7]]] == 1 &]

A385297 Numbers whose digits all belong to the same residue class mod 8.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 19, 22, 33, 44, 55, 66, 77, 80, 88, 91, 99, 111, 119, 191, 199, 222, 333, 444, 555, 666, 777, 800, 808, 880, 888, 911, 919, 991, 999, 1111, 1119, 1191, 1199, 1911, 1919, 1991, 1999, 2222, 3333, 4444, 5555, 6666, 7777, 8000, 8008, 8080, 8088, 8800, 8808, 8880, 8888
Offset: 1

Views

Author

Stefano Spezia, Jun 24 2025

Keywords

Links

Crossrefs

Similar sequences for other values of the modulo k: A059708 (k=2), A385292 (k=3), A385293 (k=4), A385294 (k=5), A385295 (k=6), A385296 (k=7), this sequence (k=8), A385298 (k=9).

Programs

  • Mathematica
    Select[Range[0,9000],Length[DeleteDuplicates[Mod[IntegerDigits[#],8]]] == 1 &]

A385298 Numbers whose digits all belong to the same residue class mod 9.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 90, 99, 111, 222, 333, 444, 555, 666, 777, 888, 900, 909, 990, 999, 1111, 2222, 3333, 4444, 5555, 6666, 7777, 8888, 9000, 9009, 9090, 9099, 9900, 9909, 9990, 9999, 11111, 22222, 33333, 44444, 55555, 66666, 77777, 88888, 90000, 90009
Offset: 1

Views

Author

Stefano Spezia, Jun 24 2025

Keywords

Links

Crossrefs

Similar sequences for other values of the modulo k: A059708 (k=2), A385292 (k=3), A385293 (k=4), A385294 (k=5), A385295 (k=6), A385296 (k=7), A385297 (k=8), this sequence (k=9).

Programs

  • Mathematica
    Select[Range[0,90000],Length[DeleteDuplicates[Mod[IntegerDigits[#],9]]] == 1 &]
Showing 1-10 of 15 results. Next

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

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