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

2,1

Comments

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

References

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

Links

Table of n, a(n) for n=2..106.

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]

Formula

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

Example

21.32746579959003668663940148693951284375095170327002181725118954... - Robert G. Wilson v, Jun 01 2009

Mathematica

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

Prog

(PARI) 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

Crossrefs

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).

Sequence in context: A144238 A319622 A348747 * A101709 A005247 A363550

Adjacent sequences: A082830 A082831 A082832 * A082834 A082835 A082836

Keyword

nonn,cons,base

Author

Robert G. Wilson v, Apr 14 2003

Extensions

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

Status

approved