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.

Previous Showing 11-15 of 15 results.

A179951 Decimal expansion of Sum_{k has exactly two bits equal to 1 in base 2} 1/k.

Original entry on oeis.org

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

Views

Author

Robert G. Wilson v, Aug 03 2010

Keywords

Comments

Obviously for k > 0 in base 2 having no bit equal to 1 the sum is 0 and for 1 bit equal to 1 the sum is 2.

Examples

			Sum_{k>0} 1/A018900(k) = 1.52899956069688841838263949451...

Links

Crossrefs

Cf. A065442, A082830, A082831, A082832, A082833, A082834, A082835, A082836, A082837, A082838, A082839, A140502, A160502, A018900, A323482.

Programs

  • Maple
     evalf( 2*add( (-1)^(n+1)*((4^n + 1)/(4^n - 1))*(1/2)^(n^2), n = 1..18), 100); # Peter Bala, Jan 28 2022
  • Mathematica
    (* first install irwinSums.m, see either reference, then *) First@ RealDigits@ iSum[1, 2, 2^7, 2]

Formula

Equals Sum_{j>=1} Sum_{i=0..j-1} 1/(2^i + 2^j).
From Amiram Eldar, Jun 30 2020: (Start)
Equals Sum_{k>=0} 1/(2^k + 1/2).
Equals 2 * A323482 - 1. (End)
Equals 2*Sum_{n >= 1} (-1)^(n+1)*((4^n + 1)/(4^n - 1))*(1/2)^(n^2). The first 18 terms of the series gives the constant correct to more than 100 decimal places. - Peter Bala, Jan 28 2022

A375805 Decimal expansion of Sum_{n >= 1} 1/A171397(n).

Original entry on oeis.org

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

Views

Author

Robert C. Lyons, Aug 29 2024

Keywords

Comments

A variation on the harmonic series, in which the denominators are treated as base 11 numbers. Equivalently: sum of reciprocals of positive integers whose base-11 representation contains no digit A (no "10" digit).
Values were calculated using Mathematica code from Baillie & Schmelzer (see link). Note that the code in the Wolfram Library Archive, as it stands, does not support digits > 9 in bases > 10 (and doing the "obvious" thing will be interpreted as asking a different question with a different answer); the code was modified to support this.
Kempner (1914) showed that this series converges. - N. J. A. Sloane, Aug 31 2024
There is a slight ambiguity when we get to 1/10. This is to be regarded as 1/(1*11 + 0*1) = (1/11)-in-base-10 and not as 1/A = 1/(10*1) = (1/10)-in-base-10. - N. J. A. Sloane, Aug 30 2024

Examples

			26.2833282048814207699401516874442229241887980925...

References

  • Burnol, Jean-François. "Moments in the exact summation of the curious series of Kempner type." arXiv preprint arXiv:2402.08525 (2024).
  • A. J. Kempner, A Curious Convergent Series, American Mathematical Monthly, 21 (February, 1914), pp. 48-50. (https://dx.doi.org/10.2307/2972074)
  • Schmelzer, Thomas, and Robert Baillie. "Summing a curious, slowly convergent series." The American Mathematical Monthly 115.6 (2008): 525-540.

Links

Crossrefs

Cf. A171397, A375523/A375524.
See also A082830, A082831, A082832, A082833, A082834, A082835, A082836, A082837, A082838, A082839.

Extensions

Corrected data provided by Gareth McCaughan, Sep 02 2024

A179954 Decimal expansion of the sum of the reciprocals of pandigital numbers in which each digit appears exactly once.

Original entry on oeis.org

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

Views

Author

Robert G. Wilson v, Aug 03 2010

Keywords

Comments

This is example in 3. 1(a) of R. Baillie, revised.
This is a finite sum so it is a rational number.

Examples

			0.0008258903479192529386079577501789135432537929965887385729771528345968177...

Links

Crossrefs

Cf. A050278, A010784, A082830-A082839, A140502, A160502.

Formula

Sum_{k=1..3265920} 1/A050278(k).

Extensions

Standardized offset and leading zeros from R. J. Mathar, Aug 06 2010
More terms from Robert G. Wilson v, Sep 07 2010

A194183 Decimal expansion of the (finite) value of Sum_{ k>=1, k has no "0" digit in base 100 } 1/k.

Original entry on oeis.org

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

Views

Author

Robert G. Wilson v, Aug 18 2011

Keywords

Examples

			460.525202638512471142936753566415294971256909908479340601695672872...

Links

Crossrefs

Cf. A082839.

Programs

  • Mathematica
    RealDigits[kSum[0, 120, 100]][[1]] (* Amiram Eldar, Jun 15 2023, using Baillie and Schmelzer's kempnerSums.nb, see Links *)

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

Views

Author

Amiram Eldar, Oct 20 2020

Keywords

Comments

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).

Examples

			65.74331110185328196734583167680868411685344106635398...

Links

  • 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.

Crossrefs

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).

Formula

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.
Previous Showing 11-15 of 15 results.

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