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.

Showing 1-3 of 3 results.

A194368 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) = 0, where r=sqrt(2) and < > denotes fractional part.

Original entry on oeis.org

2, 4, 12, 14, 16, 24, 26, 28, 70, 72, 74, 82, 84, 86, 94, 96, 98, 140, 142, 144, 152, 154, 156, 164, 166, 168, 408, 410, 412, 420, 422, 424, 432, 434, 436, 478, 480, 482, 490, 492, 494, 502, 504, 506, 548, 550, 552, 560, 562, 564, 572, 574, 576, 816, 818
Offset: 1

Views

Author

Clark Kimberling, Aug 23 2011

Keywords

Comments

Suppose that r and c are real numbers, 0 < c < 1, and
...
s(m) = Sum_{k=1..m} ( - )
...
where < > denotes fractional part. The inequalities s(m) < 0, s(m) = 0, s(m) > 0 yield up to three sequences that partition the set of positive integers, as in the examples cited below. Of particular interest are choices of r and c for which s(m) >= 0 for every m >= 1.
.
Note that s(m) = m*c - Sum_{k=1..m} floor(c + ). This shows that if c is a rational number p/q, then the range of s(m) is a set of rational numbers having denominator q. In this case, it is easy to prove that if s(m)=0, then m is an integer multiple of q, yielding a sequence of quotients denoted by [[m/q>]] in the following list:
.
r..........p/q....s(m)<0....s(m)=0....[[m/q]]...s(m)>0
sqrt(2)....1/2....(empty)...A194368...A194369...A194370
sqrt(3)....1/2....A194371...A194372.............A194373
sqrt(5)....1/2....(empty)...A194374.............A194375
sqrt(6)....1/2....(empty)...A194376.............A194377
sqrt(7)....1/2....A194378...A194379.............A194380
sqrt(8)....1/2....A194381...A194382...A194383...A194384
sqrt(10)...1/2....(empty)...A194385.............A194386
sqrt(11)...1/2....A194387...A194388.............A194389
sqrt(12)...1/2....(empty)...A194390.............A194391
sqrt(13)...1/2....A194392...A194393.............A194394
sqrt(14)...1/2....A194395...A194396.............A194397
sqrt(15)...1/2....A194398...A194399.............A194400
tau........1/2....A194401...A194402...A194403...A194404
e..........1/2....A194405...A194406.............A194407
Pi.........1/2....A194408...A194409.............A194410
sqrt(2)....1/3....A194411...A194412...A194413...A194414
sqrt(3)....1/3....A194415...A194416...A194417...A194418
sqrt(5)....1/3....A194419...A194420.............A194421
sqrt(2)....2/3....A194422...A194423...A194424...A194425
tau...../2...A194461.......................A194462
tau........A194463.......................A194464
sqrt(2)....1/r.......A194465....................A194466
sqrt(3)....1/r.......A194467....................A194468
.
Next, suppose that r and c are chosen so that s(m)=0 for all m. Then the sets X={m : s(m)<0} and Y={m : s(m)>0} represent a pair of "generalized Beatty sequences" in this sense: if c=1/, the sets X and Y represent the Beatty sequences of 1/ and 1<-r>. Examples:
...
r..........c.........X.........Y......
sqrt(2)....r-1.......A003151...A003152
sqrt(3)....r-1.......A003511...A003512
tau........r-1.......A000201...A001950
sqrt(1/2)..r.........A001951...A001952
e..........e-2.......A000062...A098005

References

  • Ivan Niven, Diophantine Approximations, Interscience Publishers, 1963.

Links

Crossrefs

Cf. A184369=(1/2)*A184368, A194285, A194469, A194470.

Programs

  • Mathematica
    r = Sqrt[2]; c = 1/2;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t1, 1]] (* empty *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 800}];
Flatten[Position[t2, 1]] (* A194368 *)
%/2 (* A194369 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t3, 1]] (* A194370 *)

A248910 Numbers with no zeros in base-6 representation.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 91, 92
Offset: 1

Views

Author

Reinhard Zumkeller, Mar 08 2015

Keywords

Comments

Different from A039215, A047253, A184522, A187390, A194386.

Links

Crossrefs

Cf. A007092, A100969 (subsequence).
Zeroless numbers in some other bases <= 10: A000042 (base 2), A032924 (base 3), A023705 (base 4), A255805 (base 8), A255808 (base 9), A052382 (base 10).

Programs

  • Haskell
    a248910 n = a248910_list !! (n-1)
a248910_list = iterate f 1 where
   f x = 1 + if r < 5 then x else 6 * f x'  where (x', r) = divMod x 6
  • Mathematica
    Select[Range[100], DigitCount[#,6, 0] == 0 &] (* Paolo Xausa, Jun 29 2025 *)
  • PARI
    isok(m) = vecmin(digits(m, 6)) > 0; \\ Michel Marcus, Jan 23 2022
  • Python
    from sympy import integer_log
def A248910(n):
    m = integer_log(k:=(n<<2)+1,5)[0]
    return sum((1+(k-5**m)//(5**j<<2)%5)*6**j for j in range(m)) # Chai Wah Wu, Jun 28 2025

A194385 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) = 0, where r=sqrt(10) and < > denotes fractional part.

Original entry on oeis.org

6, 12, 18, 24, 30, 36, 228, 234, 240, 246, 252, 258, 264, 456, 462, 468, 474, 480, 486, 492, 684, 690, 696, 702, 708, 714, 720
Offset: 1

Views

Author

Clark Kimberling, Aug 23 2011

Keywords

Comments

See A194368.

Crossrefs

Cf. A010467, A194368, A194386.

Programs

  • Mathematica
    r = Sqrt[10]; c = 1/2;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t1, 1]]  (* empty *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 800}];
Flatten[Position[t2, 1]]     (* A194385 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t3, 1]]     (* A194386 *)
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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