A052406 -id:A052406 - cp's OEIS frontend

A082833 Decimal expansion of Kempner series Sum_{k >= 1, k has no digit 4 in base 10} 1/k.

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

Offset: 2

Robert G. Wilson v, Apr 14 2003

Numbers with a digit 4 (A011534) 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

			21.32746579959003668663940148693951284375095170327002181725118954... - _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, A052406 (numbers with no 4), A011534 (numbers with a 4).

Cf. A082830, A082831, A082832, A082834, 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 *)

sumpos(k=2,1/A052406(k)) \\ For illustration only, slow and not very precise: with \p19 takes 2 sec to get 5 digits right. - M. F. Hasler, Jan 13 2020

Equals Sum_{k in A052406\{0}} 1/k, where A052406 = numbers with no digit 3. - M. F. Hasler, Jan 15 2020

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

A089590 Uban numbers (the letter u is banned from the English name of the number).

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, 93, 95, 96, 97, 98, 99

Offset: 1

Eric W. Weisstein, Nov 09 2003

The sequence of uban numbers first differs from A052406 (the numbers not containing the digit 4) at the term 40 (forty), which is a uban number but is not 4-less.

Daniel Arribas and Reinhard Zumkeller, Table of n, a(n) for n = 1..1000 (first 89 terms from Daniel Arribas)
Roel and Bas van Dijk, Numerals package, Hackage (Haskell packages)
Eric Weisstein's World of Mathematics, Uban Number
Index entries for sequences related to number of letters in n

Cf. A052406.

Cf. A006933 (ban e), A089589 (ban i), A008521 (ban o), A008523 (ban t).

import Data.Maybe (fromJust)
import Data.Text (Text); import qualified Data.Text as T (all)
import Text.Numeral.Grammar.Reified (defaultInflection)
import qualified Text.Numeral.Language.EN as EN  -- see link
a089590 n = a089590_list !! (n-1)
a089590_list = filter (T.all (/= 'u') . numeral) [0..] where
   numeral :: Integer -> Text
   numeral = fromJust . EN.gb_cardinal defaultInflection
-- Reinhard Zumkeller, Jan 23 2015

from num2words import num2words
from itertools import islice, product
def ok(n): return "u" not in num2words(n)
def agen(): # generator of terms < 10**304
    base, pows = [k for k in range(1, 1000) if ok(k)], [1]
    yield from ([0] if ok(0) else []) + base
    for e in range(3, 304, 3):
        if "u" not in num2words(10**e)[4:]:
            pows = [10**e] + pows
            for t in product([0] + base, repeat=len(pows)):
                if t[0] == 0: continue
                yield sum(t[i]*pows[i] for i in range(len(t)))
print(list(islice(agen(), 100))) # Michael S. Branicky, Aug 19 2022

a(1) = 0 prepended by Reinhard Zumkeller, Jan 23 2015

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.

A352735 Lucky numbers in Chinese: Numbers whose decimal expansion contains 8 but not 4.

Original entry on oeis.org

8, 18, 28, 38, 58, 68, 78, 80, 81, 82, 83, 85, 86, 87, 88, 89, 98, 108, 118, 128, 138, 158, 168, 178, 180, 181, 182, 183, 185, 186, 187, 188, 189, 198, 208, 218, 228, 238, 258, 268, 278, 280, 281, 282, 283, 285, 286, 287, 288, 289, 298, 308, 318, 328, 338, 358

Offset: 1

Charles R Greathouse IV, Mar 30 2022

The number 8 (八) sounds like 发 = "to get rich". The number 4 (四) sounds like 死 = "to die". Many numbers are auspicious or inauspicious in Chinese numerology but these are perhaps the most common.

Intersection of A052406 and A011538.

is(n)=setintersect(Set(digits(n)),[4,8])==[8]

a(n) ≍ n^k where k = A154160 = 1.047951637....

cp's OEIS Frontend

A082833 Decimal expansion of Kempner series Sum_{k >= 1, k has no digit 4 in base 10} 1/k.

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A089590 Uban numbers (the letter u is banned from the English name of the number).

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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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Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A352735 Lucky numbers in Chinese: Numbers whose decimal expansion contains 8 but not 4.

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Author

Keywords

Comments

Crossrefs

Programs

PARI

Formula