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 1-4 of 4 results.

A342726 Niven numbers in base i-1: numbers that are divisible by the sum of their digits in base i-1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 12, 15, 16, 18, 20, 24, 25, 30, 32, 33, 35, 36, 40, 42, 44, 45, 48, 50, 54, 60, 64, 65, 66, 70, 77, 80, 88, 90, 96, 99, 100, 110, 112, 120, 124, 125, 126, 130, 140, 144, 145, 147, 150, 156, 160, 168, 170, 180, 182, 184, 185, 186, 190, 192
Offset: 1

Views

Author

Amiram Eldar, Mar 19 2021

Keywords

Comments

Numbers k that are divisible by A066323(k).
Equivalently, Niven numbers in base -4, since A066323(k) is also the sum of the digits of k in base -4.

Examples

			2 is a term since its representation in base i-1 is 1100 and 1+1+0+0 = 2 is a divisor of 2.
10 is a term since its representation in base i-1 is 111001100 and 1+1+1+0+0+1+1+0+0 = 5 is a divisor of 10.

Links

Crossrefs

Cf. A007608, A066321, A066323, A271472, A342725, A342727, A342728, A342729.
Similar sequences: A005349 (decimal), A049445 (binary), A064150 (ternary), A064438 (quaternary), A064481 (base 5), A118363 (factorial), A328208 (Zeckendorf), A328212 (lazy Fibonacci), A331085 (negaFibonacci), A333426 (primorial), A334308 (base phi), A331728 (negabinary), A342426 (base 3/2).

Programs

  • Mathematica
    v = {{0, 0, 0, 0}, {0, 0, 0, 1}, {1, 1, 0, 0}, {1, 1, 0, 1}}; q[n_] := Divisible[n, Total[Flatten @ v[[1 + Reverse @ Most[Mod[NestWhileList[(# - Mod[#, 4])/-4 &, n, # != 0 &], 4]]]]]]; Select[Range[200], q]

A342729 Self numbers in base i-1: numbers not of the form k + A066323(k).

Original entry on oeis.org

1, 3, 5, 7, 9, 22, 24, 26, 39, 41, 43, 56, 58, 60, 73, 75, 77, 90, 92, 94, 107, 109, 111, 136, 138, 140, 153, 155, 157, 170, 172, 174, 199, 201, 203, 216, 218, 220, 233, 235, 237, 262, 264, 266, 279, 281, 283, 296, 298, 300, 313, 315, 317, 330, 332, 334, 347, 349
Offset: 1

Views

Author

Amiram Eldar, Mar 19 2021

Keywords

Comments

Equivalently, self numbers in base -4, since A066323(k) is also the sum of the digits of k in base -4.
Analogous to self numbers (A003052) using base i-1 representation (A271472) instead of decimal expansion.
The number of terms not exceeding 10^k, for k=1,2,..., is 5, 20, 155, 1507, 15008, 150007, 1500014, 15000011. Is the asymptotic density of this sequence exactly 3/20?

References

  • József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384-386.

Links

Crossrefs

Cf. A007608, A066323, A066321, A271472, A342725, A342726, A342727, A342728.
Similar sequences: A003052 (decimal), A010061 (binary), A010064 (base 4), A010067 (base 6), A010070 (base 8), A339211 (Zeckendorf), A339212 (dual Zeckendorf), A339213 (base phi), A339214 (factorial base), A339215 (primorial base).

Programs

  • Mathematica
    s[n_] := Module[{v = {{0, 0, 0, 0}, {0, 0, 0, 1}, {1, 1, 0, 0}, {1, 1, 0, 1}}}, Plus @@ Flatten @ v[[1 + Reverse @ Most[Mod[NestWhileList[(# - Mod[#, 4])/-4 &, n, # != 0 &], 4]]]]]; f[n_] := n + s[n]; m = 1000; Complement[Range[m], Select[Union@Array[f, m], # <= m &]]

A342727 Digitally balanced numbers in base i-1: numbers that in base i-1 have the same number of 0's as 1's.

Original entry on oeis.org

2, 21, 26, 31, 36, 41, 46, 51, 310, 315, 325, 330, 335, 340, 345, 350, 355, 360, 365, 370, 375, 390, 395, 405, 410, 415, 420, 425, 430, 435, 455, 470, 475, 485, 490, 495, 535, 550, 555, 565, 570, 575, 580, 585, 590, 595, 600, 605, 610, 620, 625, 630, 635, 645
Offset: 1

Views

Author

Amiram Eldar, Mar 19 2021

Keywords

Examples

			2 is a term since its representation in base i-1, 1100, has 2 0's and 2 1's.
21 is a term since its representation in base i-1, 110011010001, has 6 0's and 6 1's.

Links

Crossrefs

Cf. A066321, A066323, A271472, A342725, A342726, A342728, A342729.
Similar sequences: A031443 (binary), A210619 (Zeckendorf).

Programs

  • Mathematica
    v = {{0, 0, 0, 0}, {0, 0, 0, 1}, {1, 1, 0, 0}, {1, 1, 0, 1}}; balQ[n_] := Plus @@ (d = IntegerDigits[n]) == Length[d]/2; q[n_] := balQ @ FromDigits[Flatten@v[[1 + Reverse @ Most[Mod[NestWhileList[(# - Mod[#, 4])/-4 &, n, # != 0 &], 4]]]]]; Select[Range[1000], q]

A342728 a(n) is the least number k such that A066323(k) = n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 23, 39, 55, 71, 87, 103, 359, 615, 871, 1127, 1383, 1639, 5735, 9831, 13927, 18023, 22119, 26215, 91751, 157287, 222823, 288359, 353895, 419431, 1468007, 2516583, 3565159, 4613735, 5662311, 6710887, 23488103, 40265319, 57042535, 73819751
Offset: 0

Views

Author

Amiram Eldar, Mar 19 2021

Keywords

Comments

a(n) is the least number k whose sum of digits in base i-1 (or in base -4) is n.

Links

Crossrefs

Cf. A007608, A066321, A066323, A271472, A342725, A342726, A342727, A342729.

Programs

  • Mathematica
    Join[{0}, LinearRecurrence[{1,0,0,0,0,16,-16}, Range[7], 50]]

Formula

a(n) = n for n <= 7, and a(n) = a(n-1) + 16*a(n-6) - 16*a(n-7) for n > 7.
G.f.: x*(1 + x + x^2 + x^3 + x^4 + x^5 - 15*x^6)/(1 - x - 16*x^6 + 16*x^7). - Stefano Spezia, Mar 20 2021
From Greg Dresden, Jun 21 2021: (Start)
a(3*n+1) = (24 + (4^n)*(25 - 9*(-1)^n))/40.
a(3*n+2) = (24 + (4^n)*(50 + 6*(-1)^n))/40.
a(3*n+3) = (24 + (4^n)*(75 + 21*(-1)^n))/40. (End)
Showing 1-4 of 4 results.

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