A052413 -id:A052413 - cp's OEIS frontend

A082834 Decimal expansion of Kempner series Sum_{k>=1, k has no digit 5 in base 10} 1/k.

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

Offset: 2

Robert G. Wilson v, Apr 14 2003

Numbers with a digit 5 (A011535) have asymptotic density 1, i.e., here almost all terms are removed from the harmonic series, which makes convergence less surprising. See A082839 (the analog for digit 0) for more information about such so-called Kempner series. - M. F. Hasler, Jan 13 2020

			21.83460081229691816340723504060918271784656751501391829167935918... - _Robert G. Wilson v_, Jun 01 2009

Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, page 34.

Robert Baillie, Sums of reciprocals of integers missing a given digit, Amer. Math. Monthly, 86 (1979), 372-374.
Robert Baillie, Summing the curious series of Kempner and Irwin arXiv:0806.4410 [math.CA], 2008-2015. [From _Robert G. Wilson v_, Jun 01 2009]
Wikipedia, Kempner series. [From _M. F. Hasler_, Jan 13 2020]
Wolfram Library Archive, KempnerSums.nb (8.6 KB) - Mathematica Notebook, Summing Kempner's Curious (Slowly-Convergent) Series. [From _Robert G. Wilson v_, Jun 01 2009]

Cf. A002387, A024101, A052413 (numbers with no '5'), A011535 (numbers with a '5').

Cf. A082830, A082831, A082832, A082833, A082835, A082836, A082837, A082838, A082839 (analog for digits 1, 2, ..., 9 and 0).

(* see the Mmca in Wolfram Library Archive. - Robert G. Wilson v, Jun 01 2009 *)

Equals Sum_{k in A052413\{0}} 1/k, where A052413 = numbers with no digit 5. - M. F. Hasler, Jan 15 2020

More terms from Robert G. Wilson v, Jun 01 2009

Minor edits by M. F. Hasler, Jan 13 2020

A004724 Delete all 5's from the sequence of nonnegative integers.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 1, 16, 17, 18, 19, 20, 21, 22, 23, 24, 2, 26, 27, 28, 29, 30, 31, 32, 33, 34, 3, 36, 37, 38, 39, 40, 41, 42, 43, 44, 4, 46, 47, 48, 49, 0, 1, 2, 3, 4, 6, 7, 8, 9, 60, 61, 62, 63, 64, 6, 66, 67, 68, 69, 70, 71, 72, 73, 74, 7, 76

Offset: 0

N. J. A. Sloane

In contrast to the variant A004180 where a(n) = 0 when all the digits of n are 5's, here a number completely disappears in that case, so that subsequent indices are shifted and for n > 4, a(n) is not the result of deleting 5's from n: see formula. - M. F. Hasler, Jan 13 2020

			From  _M. F. Hasler_, Jan 13 2020: (Start)
After a(4) = 4 comes a(5) = 6, since the number 5 completely disappears.
a(48) = 49 is followed by 0, 1, 2, 3, 4 (i.e., 50, ..., 54 with the initial digit removed) and then a(54) = 6, because 55 disappears completely.
Illustration of the formula: as long as n < 5 (the first number that completely disappears) we have a(n) = A004180(n). Here n has 1 digit but n+1 does not exceed the (single repdigit) 5 (left hand side in the Iverson bracket), so m = 0. From n = 5 on, n+1 > 5, so m = 1.
Then, when n has L(n) = 2 digits, we still have n = 2 - 1 = 1 as long as n+2 <= 55 or n <= 53, but m = 3 for n > 55 - 2 = 53, i.e., from n = 54 on (where the term 55 has disappeared, see above).
Similarly, m = 3 for n > 555 - 3, i.e., from n >= 553 on, etc. (End)

Cf. A004180 (delete digits 5 in n), A052413 (numbers with no digit 5).

Cf. A004719, A004720, A004721, A004722, A004723, A004725, A004726, A004727, A004728.

m=1; for u=0:76 v=dec2base(u, 10)-'0'; v = v(v~=5);  if length(v)>0; sol(m)=(str2num(strrep(num2str(v), ' ', ''))); m=m+1; end; end; sol % Marius A. Burtea, Jan 16 2020

apply( {A004724(n,L=logint(n+!n,10)+1)=A004180(n+L-(10^L\9*5-L>=n))}, [0..99])
A004724_upto(N)={[fromdigits(v)|v<-[[d|d<-digits(n+!n*50),d!=5]|n<-[0..N]],#v]} \\ M. F. Hasler, Jan 13 2020

def A004724(n):
    l = len(str(n))
    m = 5*(10**l-1)//9
    k = n + l - int(n+l < m)
    return 4 if k == m else int(str(k).replace('5','')) # Chai Wah Wu, Apr 20 2021

a(n) = A004180(n + m) where m = L(n) - [ (10^L(n)-1)/9*5 >= n + L(n) ], L(n) = floor(log_10(max(n,1)) + 1), the number of digits of n, and [...] is the Iverson bracket (1 if true, 0 else). - M. F. Hasler, Jan 13 2020

A261189 Integers such that no subsequence of decimal representation is divisible by 5.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 61, 62, 63, 64, 66, 67, 68, 69, 71, 72, 73, 74, 76, 77, 78, 79, 81, 82, 83, 84, 86, 87, 88, 89, 91, 92, 93, 94, 96, 97, 98, 99, 111, 112, 113, 114, 116, 117, 118, 119, 121

Offset: 1

Zak Seidov, Aug 11 2015

Or, integers with no decimal digits 0 and 5 (see 2nd Mma).

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

Cf. A114922, A261188.

Intersection of A052382 and A052413.

import Data.List.Ordered (isect)
a261189 n = a261189_list !! (n-1)
a261189_list = a052382_list `isect` a052413_list
-- Reinhard Zumkeller, Aug 13 2015

(*1*)Reap[Do[Le=Length[id=IntegerDigits[n]];If[Min[Mod[Flatten[Table[FromDigits[Take[id,{x,y}]],{x,Le},{y,x,Le}]],5]]>0,Sow[n]],{n,199}]][[2,1]]
(*2*)Reap[Do[id=IntegerDigits[n];If[Intersection[id,{0,5}]== {},Sow[n]],{n,199}]][[2,1]]

a(n)=fromdigits(apply(d->if(d>3,d+1,d), digits(n-1,8)))+1 \\ Charles R Greathouse IV, Aug 11 2015

A052411 Number of n-crossing hyperbolic knots having symmetry group Z1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 2, 24, 173, 1047, 6709, 37177, 224311, 1301492

Offset: 1

Eric W. Weisstein

Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell., 20, 33-48, Fall 1998.

Eric Weisstein's World of Mathematics, Knot symmetry.

Cf. A052411, A052412, A052413, A052414.

A052412 Number of n-crossing hyperbolic knots having symmetry group Z2.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 14, 57, 210, 712, 2268, 7011

Offset: 1

Eric W. Weisstein

Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First 1,701,936 Knots." Math. Intell., 20, 33-48, Fall 1998.

Eric Weisstein's World of Mathematics, Knot Symmetry.

Cf. A052411, A052412, A052413, A052414.

A338287 Decimal expansion of the sum of reciprocals of the numbers that are not pandigital numbers (version 2, A171102).

Original entry on oeis.org

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

Offset: 2

Amiram Eldar, Oct 20 2020

The sum of the reciprocals of the terms of the complement of A171102: numbers with at most 9 distinct digits. It is the union of the 10 sequences of numbers without a single given digit (see the Crossrefs section).

The terms in the data section were taken from the 200 decimal digits given by Strich and Müller (2020).

			65.74331110185328196734583167680868411685344106635398...

Robert Strich and Eric Müller, Ziffernreduzierte Zahlen, in: E. Specht, E. Quaisser and P. Bauermann (eds.), 50 Jahre Bundeswettbewerb Mathematik, Springer Spektrum, Berlin, Heidelberg, 2020, pp. 171-182.

Cf. A171102, A179954, A082830, A082831, A082832, A082833, A082834, A082835, A082836, A082837, A082838, A082839.

Cf. A052382 (numbers without the digit 0), A052383 (without 1), A052404 (without 2), A052405 (without 3), A052406 (without 4), A052413 (without 5), A052414 (without 6), A052419 (without 7), A052421 (without 8), A007095 (without 9).

Equals 1/1 + 1/2 + 1/3 + ... + 1/1023456788 + 1/1023456790 + ..., i.e., A171102(1) = 1023456789 is the first number whose reciprocal is not in the sum.

cp's OEIS Frontend

A082834 Decimal expansion of Kempner series Sum_{k>=1, k has no digit 5 in base 10} 1/k.

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A004724 Delete all 5's from the sequence of nonnegative integers.

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A261189 Integers such that no subsequence of decimal representation is divisible by 5.

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A052411 Number of n-crossing hyperbolic knots having symmetry group Z1.

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A052412 Number of n-crossing hyperbolic knots having symmetry group Z2.

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A338287 Decimal expansion of the sum of reciprocals of the numbers that are not pandigital numbers (version 2, A171102).

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