Martin Janecke - cp's OEIS frontend

A285511 Value of the n-th Roman number interpreted as Latin alphabetic number.

9, 243, 6327, 256, 22, 581, 15115, 392999, 258, 24, 633, 16467, 428151, 16480, 646, 16805, 436939, 11360423, 16482, 648, 16857, 438291, 11395575, 438304, 16870, 438629, 11404363, 296513447, 438306, 16872, 438681, 11405715, 296548599, 11405728, 438694, 11406053, 296557387, 7710492071, 11405730, 636

Offset: 1

Martin Janecke, Apr 20 2017

Lists can be numbered using different counter styles, for example using the Latin alphabet A, B, C, ..., Z, AA, AB, ... or the Roman number system I, II, III, IV, V, VI, ... Both these counter styles are defined in CSS Counter Styles Level 3 as "upper-alpha" and "upper-roman". Roman number representations are defined for the range 1 to 3999 only. Roman numerals are a subset of Latin alphabet letters; for every Roman number there is exactly one alphabetic number that looks identical. Denote the n-th Roman number by R(n) and the m-th alphabetic number by L(m), then R(n) and L(a(n)) look identical.

			The number n = 1 is written "I" in the Roman number system. "I" being the ninth letter in the alphabet is also the ninth number in the alphabetic number system. Therefore a(1) = 9.
The number n = 2 is written "II" in the Roman number system. "II" is also the 243rd number in the alphabetic number system, because "I" is the ninth letter in the 26-letter alphabet and 9*26^1+9*26^0 = 243. Therefore a(2) = 243.
The number n = 3 is written "III" in the Roman number system. "III" is also the 6327th number in the alphabetic number system because "I" is the ninth letter in the 26-letter alphabet and 9*26^2+9*26^1+9*26^0 = 6327. Therefore a(3) = 6327.
The number n = 4 is written "IV" in the Roman number system. "IV" is also the 256th number in the alphabetic number system because "I" is the ninth letter in the 26-letter alphabet and "V" is the 22nd letter, therefore a(4) = 9*26^1 + 22 = 256.
The number n = 600 is written "DC" in the Roman number system. "DC" is also the 107th number in the alphabetic number system, because "D" and "C" are the fourth and third letters in the 26-letter alphabet and 4*26^1+3*26^0 = 107. Therefore a(600) = 107.

Martin Janecke, Table of n, a(n) for n = 1..3999
Tab Atkins Jr. et al., CSS Counter Styles Level 3 (W3C Candidate Recommendation), 11 June 2015
Martin Janecke, Roman numbers ↔ Latin alphabet counter, April 2017

Cf. A006968, A009970, A061493.

A281071 Largest number k such that b - r is even or r = 0 for all b = 1..k where r = n mod b.

Original entry on oeis.org

1, 2, 1, 4, 1, 6, 1, 2, 1, 10, 1, 4, 1, 2, 1, 6, 1, 6, 1, 2, 1, 4, 1, 4, 1, 2, 1, 10, 1, 6, 1, 2, 1, 4, 1, 12, 1, 2, 1, 8, 1, 4, 1, 2, 1, 6, 1, 6, 1, 2, 1, 4, 1, 4, 1, 2, 1, 6, 1, 6, 1, 2, 1, 4, 1, 14, 1, 2, 1, 10, 1, 4, 1, 2, 1, 6, 1, 8, 1, 2, 1, 4, 1, 4, 1, 2, 1, 6, 1, 6, 1, 2, 1, 4, 1, 8, 1, 2

Offset: 1

Martin Janecke, Jan 14 2017

Consider a text mode screen which is a fixed number of character columns wide. Text only breaks at the screen width, there is no manual line break. Then a(n) is the largest screen width in terms of characters so that a string of n printable characters can be perfectly centrally aligned on this and all smaller widths.

a(n) = n for n in {1, 2, 4, 6, 10}, otherwise a(n) < n. The sequence is unbounded.

			a(22) = 4 because
  22 mod 1 = 0 where r = 0,
  22 mod 2 = 0 where r = 0,
  22 mod 3 = 1 where 3 - 1 is even,
  22 mod 4 = 2 where 4 - 2 is even, but
  22 mod 5 = 2 where r > 0 and 5 - 2 is odd.

Martin Janecke, Table of n, a(n) for n = 1..10000

Cf. A281072.

a(n) = {ok = 1; k = 1; while(ok, v = vector(k, b, if ((n % b)==0, 0, b - (n%b))); ok = #select(x->((x % 2)==0), v) == k; if (ok, k++);); k--;} \\ Michel Marcus, Jan 23 2017

A281072 Smallest number k > 0 such that b - r is even or r = 0 for all b = 1..n, r == k mod b.

Original entry on oeis.org

1, 2, 4, 4, 6, 6, 10, 10, 10, 10, 36, 36, 66, 66, 196, 196, 196, 196, 568, 568, 568, 568, 1008, 1008, 1288, 1288, 1288, 1288, 8398, 8398, 8730, 8730, 50688, 50688, 124738, 124738, 124738, 124738, 155296, 155296, 155296, 155296, 155296, 155296, 155296, 155296, 1784476, 1784476

Offset: 1

Martin Janecke, Jan 14 2017

a(n) is the smallest number for which A281071(a(n)) is greater than or equal to n.

			a(3) = 4 because
  1 mod 2 = 1 where r > 0 and 2 - 1 is odd;
  2 mod 3 = 2 where r > 0 and 3 - 2 is odd;
  3 mod 2 = 1 where r > 0 and 2 - 1 is odd;
  4 mod 1 = 0 where r = 0,
  4 mod 2 = 0 where r = 0,
  4 mod 3 = 1 where 3 - 1 is even.

Martin Janecke, Table of n, a(n) for n = 1..88

Cf. A281071.

A263207 Number of integer solutions for Product_{k=1..n}(c(k) + 1) = 2 * Product_{k=1..n}(c(k) - 1) with 1 < c(k) <= c(k+1).

Original entry on oeis.org

0, 1, 2, 20, 374, 21313, 5115140

Offset: 0

Martin Janecke, Oct 12 2015

Geometrically interpreted, the sequence a(n) provides the number of distinct ways to cut an n-dimensional cube orthogonally into equally many outer parts, i.e., those that can be seen from the outside, and inner parts. All cuts must go through the whole body. c(k) is the number of cuts for the k-th dimension.
See section Example for all solutions for n=1 and n=2 and section Links for all solutions for n=3, n=4, n=5.
For any n>=1 the solution given by c(k)=A204321(k) for k=1..n-1 and c(n)=A204321(n)-1 always exists. Conjecture: there is no solution with a greater c(n). - Martin Janecke, Dec 01 2015
From Wolfdieter Lang, Dec 01 2015: (Start)
In terms of the j-th elementary symmetric functions sigma(n, j) in the indeterminates [c(1), ..., c[n]] the equation with the products can be rewritten as Sum_{j=0..n} ((1 - 2*(-1)^(n-j))*sigma(n, j) = 0, with sigma(n, 0) = 1.
If one uses the reciprocals x(k) = 1/c(k) then
  0 < x(k+1) <= x(k) <  1 (the c's are all >1, and finite), and the original equation becomes (by taking logarithms)
  Sum_{k=1..n} arctanh(x(k)) = log(2)/2 = arctanh(1/3), (which  is about .35).
  Here only positive terms appear. It is clear from the monotony of arctanh that all the x(k) < 1/3 for n >= 2, hence c(1) >= 4 for n >= 2.
(End)

			a(1) = 1, because there is exactly one solution: c(1) = 3.
a(2) = 2, because there are two solutions: c(1) = 4, c(2) = 11 and c(1) = 5, c(2) = 7. The first solution can be illustrated geometrically as a square cut into thirty outer and equally many inner parts:
    OOOOOOOOOOOO
    OIIIIIIIIIIO
    OIIIIIIIIIIO
    OIIIIIIIIIIO
    OOOOOOOOOOOO,
  The second solution yields twenty-four parts of each kind:
    OOOOOOOO
    OIIIIIIO
    OIIIIIIO
    OIIIIIIO
    OIIIIIIO
    OOOOOOOO.

M. Janecke, Kuchengerechtigkeit in n Dimensionen, includes a Python program that finds solutions for a given n, July 2015.
M. Janecke, On a conjecture about cutting hypercubes, Nov 2015.
Martin Janecke, All solutions for n=3
Martin Janecke, All solutions for n=4
Martin Janecke, All solutions for n=5

Cf. A204321.

A263151 Decimal expansion of the ratio of the length of the latus rectum arc of any parabola to its focal length: sqrt(8) + log(3 + sqrt(8)).

Original entry on oeis.org

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

Offset: 1

Martin Janecke, Oct 11 2015

Twice the universal parabolic constant A103710.

			4.591174298785276148068596098378980775195664407277166967859950693...

Index entries for transcendental numbers

Equals twice A103710. Equals A010466 + A244920.

First@ RealDigits[N[# + Log[3 + #] &@ Sqrt@ 8, 102]] (* Michael De Vlieger, Oct 11 2015 *)

sqrt(8) + log(3 + sqrt(8)) \\ Michel Marcus, Oct 11 2015

A194807 Decimal expansion of 1/(e-2).

Original entry on oeis.org

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

Offset: 1

Martin Janecke, May 06 2012

The value of the continued fraction 1+1/(2+2/(3+3/(4+4/(5+5/(6+6/(...)))))).

			1.392211191177332814376552878479816528373978385315...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for transcendental numbers

Cf. A073333 (1/(e-1)), A002627, A185108.

1/(Exp(1) - 2); // G. C. Greubel, Apr 09 2018

RealDigits[1/(E - 2), 10, 105][[1]] (* T. D. Noe, May 07 2012 *)
Fold[Function[#2 + #2/#1], 1, Reverse[Range[100]]] // N[#, 105]& // RealDigits // First (* Jean-François Alcover, Sep 19 2014 *)

default(realprecision,110);
1/(exp(1)-2)
\\ Joerg Arndt, May 07 2012

Define s(n) = Sum_{k = 2..n} 1/k! for n >= 2. Then 1/(e - 2) = 2! - Sum_ {n >= 2} 1/( (n+1)!*s(n)*s(n+1) ) is a rapidly converging series of rationals. Cf. A073333. Equivalently, 1/(e - 2) = 2! - 2!/(1*4) - 3!/(4*17) - 4!/(17*86) - ..., where [1, 4, 17, 86, ... ] is A056542. Cf. A002627 and A185108. - Peter Bala, Oct 09 2013

A205325 Decimal expansion of the limit of [0;1,1,...] + [0;2,2,...] + ... + [0;n,n,...] - log(n) as n approaches infinity.

Original entry on oeis.org

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

Offset: 0

Martin Janecke, Jan 26 2012

			0.0416662....

Martin Janecke, Edle Reihe

Cf. A001620, A205326, continued fractions A001622, A014176, A098316, A098317, A098318.

digits = 10; dn = 1000000; Clear[f]; f[n_] := NSum[2/(k + Sqrt[k^2+4]) - 1/k, {k, 1, Infinity}, NSumTerms -> 200000, WorkingPrecision -> digits+10, Method -> {"EulerMaclaurin", Method -> {"NIntegrate", "MaxRecursion" -> 20}}] + EulerGamma // RealDigits[#, 10, digits+2]& // First; f[dn]; f[n = 2*dn]; While[f[n] != f[n-dn], n = n+dn]; Prepend[ f[n][[1 ;; digits]], 0] (* Jean-François Alcover, Feb 25 2013 *)

lim_{n->infinity} (1/[1;1,...] + 1/[2;2,...] + 1/[3;3,...] + ... + 1/[n;n,...] - log(n)).

lim_{n->infinity} (sum_{k=1...n} (2/(k + sqrt(k^2 + 4))) - log(n)).

More terms from Jean-François Alcover, Feb 25 2013

More terms from Jon E. Schoenfield, Jan 05 2014

A205326 Decimal expansion of the sum of [0;n,n,n,...]^2 for n=1..infinity.

Original entry on oeis.org

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

Offset: 0

Martin Janecke, Jan 26 2012

This is the total area of all squares with sides parallel to the axes of the Cartesian coordinate system, the lower left vertex at (n,0) and the upper right vertex on f(x)=1/x for n=1..infinity.

			0.9155879199018197251998168538031900897353...

Martin Janecke, Edle Reihe

Cf. A013661, A205325, continued fractions A001622, A014176, A098316, A098317, A098318.

zeta(2)+sumpos(n=1,4/(n+sqrt(n^2+4))^2-1/n^2) \\ Charles R Greathouse IV, Jan 26 2012

Sum_{n>=1} 1/[n;n,n,...]^2.

Sum_{n>=1} 4/(n + sqrt(n^2 + 4))^2.

a(-5)-a(-86) from Charles R Greathouse IV, Jan 26 2012

cp's OEIS Frontend

User: Martin Janecke

A285511 Value of the n-th Roman number interpreted as Latin alphabetic number.

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A281071 Largest number k such that b - r is even or r = 0 for all b = 1..k where r = n mod b.

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A281072 Smallest number k > 0 such that b - r is even or r = 0 for all b = 1..n, r == k mod b.

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A263207 Number of integer solutions for Product_{k=1..n}(c(k) + 1) = 2 * Product_{k=1..n}(c(k) - 1) with 1 < c(k) <= c(k+1).

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A263151 Decimal expansion of the ratio of the length of the latus rectum arc of any parabola to its focal length: sqrt(8) + log(3 + sqrt(8)).

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A194807 Decimal expansion of 1/(e-2).

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Magma

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A205325 Decimal expansion of the limit of [0;1,1,...] + [0;2,2,...] + ... + [0;n,n,...] - log(n) as n approaches infinity.

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A205326 Decimal expansion of the sum of [0;n,n,n,...]^2 for n=1..infinity.

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