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.

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A082831 Decimal expansion of Sum_{k >= 1, k has no digit 2 in base 10} 1/k.

Original entry on oeis.org

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

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Author

Robert G. Wilson v, Apr 14 2003

Keywords

Comments

Such sums are called Kempner series, see A082839 (analog for digit 0) for more information. - M. F. Hasler, Jan 13 2020

Examples

			19.25735653280807222453277677019445411552605383115487014986836294...

References

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

Links

Crossrefs

Cf. A002387, A024101, A052404 (numbers with no digit 2).
Cf. A082830, A082832, A082833, A082834, A082835, A082836, A082837, A082838, A082839 (analog for digits 1, 3, 4, ..., 9 and 0).

Programs

  • Mathematica
    (* see the Mmca in Wolfram Library Archive *)

Formula

Equals Sum_{k in A052404\{0}} 1/k, where A052404 = numbers with no digit 2: these are omitted in the harmonic series. - M. F. Hasler, Jan 13 2020

Extensions

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

A285470 Numbers k where "2" appears as the second digit of the decimal representation.

Original entry on oeis.org

12, 22, 32, 42, 52, 62, 72, 82, 92, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 620, 621, 622, 623, 624, 625, 626, 627
Offset: 1

Views

Author

Jamie Robert Creasey, Apr 19 2017

Keywords

Comments

To find a(n), concatenate the first digit of n with 2 and then the other digits (if any) from n. See example. - David A. Corneth, Jun 12 2017

Examples

			a(21) = 221, a(36) = 326.
As the first digit of 983 is 9, and the others are 83, a(983) = 9283. - _David A. Corneth_, Jun 12 2017

Links

Crossrefs

Cf. A011532 (containing 2), A052404 (without 2), A217394 (starting with 2).

Programs

  • Maple
    seq(seq(seq(a*10^d + 2*10^(d-1)+c, c=0..10^(d-1)-1),a=1..9),d=1..2); # Robert Israel, Jun 12 2017
  • Mathematica
    Table[FromDigits@ Apply[Join, {{First@ #}, {2}, Rest@ #}] &@ IntegerDigits@ n, {n, 67}] (* Michael De Vlieger, Jun 12 2017 *)
Select[Range[700],NumberDigit[#,IntegerLength[#]-2]==2&] (* Harvey P. Dale, Aug 15 2025 *)
  • PARI
    isok(n) = (n>9) && digits(n)[2] == 2; \\ Michel Marcus, Jun 12 2017
  • PARI
    a(n) = my(d = digits(n)); fromdigits(concat([d[1], [2], vector(#d-1, i, d[i+1])])) \\ David A. Corneth, Jun 12 2017
  • PARI
    nxt(n) = if(isok(n+1), n+1, d = digits(n); t = 9*10^(#d-2); if(d[1]==9,t*=3); n+=t++) \\ David A. Corneth, Jun 12 2017
  • Python
    def a(n): s = str(n); return int(s[0] + "2" + s[1:])
print([a(n) for n in range(1, 68)]) # Michael S. Branicky, Dec 22 2021

Formula

From Robert Israel, Jun 12 2017: (Start)
a(10*n+j) = 10*a(n)+j for 0<=j<=9 and n >= 1.
G.f. g(x) satisfies g(x) = 10*(1-x^10)*g(x^10)/(1-x) + (x + 2*x + ... + 9*x^9)*x^10/(1-x^10) + 12*x + 22*x^2 + ... + 92*x^9. (End)

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 11-13 of 13 results.

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