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.

A260593 The values of the modified Syracuse algorithm, msa, in the order in which they appear in A260590.

Original entry on oeis.org

4, 2, 7, 5, 59, 56, 8, 54, 51, 45, 42, 31, 15, 40, 21, 29, 13, 12, 20, 27, 24, 10, 16, 18, 23, 39, 81, 35, 37, 26, 80, 34, 78, 43, 32, 61, 58, 50, 48, 46, 70, 65, 69, 53, 64, 77, 73, 72, 105, 75, 67, 83, 62, 92, 135, 126, 86, 111, 129, 124, 123, 127, 88, 119, 108, 100
Offset: 1

Views

Author

Joseph K. Horn and Robert G. Wilson v, Aug 30 2015

Keywords

Comments

See A260590 for the definition of the msa.
Sorted: 2, 4, 5, 7, 8, 10, 12, 13, 15, 16, 18, 20, 21, 23, 24, 26, 27, 29, 31, 32, 34, 35, 37, 39, 40, 42, 43, 45, 46, 48, 50, 51, 53, 54, 56, 58, 59, 61, 62, 64, 65, 67, 69, 70, 72, 73, 75, 77, 78, 80, 81, 83, 85, 86, 88, 89, 91, 92, 94, 96, 97, 99, 100, ... (A020914(n) for n>0).
Record values: 4, 7, 59, 81, 105, 135, 164, 165, 173, 176, 183, 224, 246, 287, 292, 298, 308, 376, 395, 398, 433, 447, ... .
Record last values to appear: 2, 5, 8, 10, 16, 18, 23, 26, 32, 46, 53, 62, 85, 94, 99, 102, 107, 115, 118, 130, 132, 134, 148, ... .

Examples

			Every odd number greater than 1 yields a msa value. a(1) is 4 and it corresponds to A260590(1).
a(2) is 2 since A260590(2) is 2.
a(3) is 7 since A260590(3) is 7.
a(4) is 5 since A260590(5) is 5, A260590(4) is 2 but it already appears as a(2).

Crossrefs

Cf. A260590, A260594.

Programs

  • Mathematica
    msa[n_] := If[ OddQ@ n, (3n + 1)/2, n/2]; f[n_] := Block[{k = 2n + 1}, Length@ NestWhileList[ msa@# &, k, # >= k &] - 1]; k = 1; lst = {}; While[k < 10000001, a = f@ k; If[ !MemberQ[lst, a], AppendTo[lst, a]]; k++]; lst

A260594 a(n) is the least k such that A260590(k) = A260593(n).

Original entry on oeis.org

1, 2, 3, 5, 13, 15, 19, 23, 35, 45, 51, 55, 63, 77, 79, 83, 95, 115, 119, 125, 141, 143, 179, 255, 319, 335, 351, 435, 463, 495, 527, 653, 791, 819, 847, 909, 1023, 1143, 1699, 1715, 2127, 2295, 3191, 3197, 3443, 3639, 3763, 3981, 5043, 6199, 6719, 7519, 8845
Offset: 1

Views

Author

Joseph K. Horn and Robert G. Wilson v, Aug 30 2015

Keywords

Comments

This is the index corresponding to, or responsible for, A260593.

Examples

			13 is the least k such that A260590(k) = 59 = A260593(5), so a(5) = 13.

Crossrefs

Cf. A260590, A260593.

Programs

  • Mathematica
    msa[n_] := If[ OddQ@ n, (3n + 1)/2, n/2]; f[n_] := Block[{k = 2n + 1}, Length@ NestWhileList[ msa@# &, k, # >= k &] - 1]; k = 1; lst = lsu = {}; While[k < 10000001, a = f@ k; If[ !MemberQ[lst, a], AppendTo[lst, a]; AppendTo[lsu, k]]; k++]; lsu

A260591 a(n) is the number of odd numbers k < 2^n such that A260590(k) = n.

Original entry on oeis.org

0, 1, 0, 1, 2, 0, 3, 7, 0, 12, 0, 30, 85, 0, 173, 476, 0, 961, 0, 2652, 8045, 0, 17637, 51033, 0, 108950, 312455, 0, 663535, 0, 1900470, 5936673, 0, 13472296, 39993895, 0, 87986917, 0, 257978502, 820236724, 0, 1899474678, 5723030586, 0, 12809477536, 38036848410, 0, 84141805077, 0, 248369601964
Offset: 1

Views

Author

Joseph K. Horn, O. Praem, and Robert G. Wilson v, Jul 29 2015

Keywords

Comments

a(n) is either 0 or about c^(n-1) with c = log(3)/log(2).
Nonzero values give A100982. - Ruud H.G. van Tol, Nov 25 2021
A close variant of this sequence, that starts at offset 0, but with a(0)=0 and a(1)=1, maps it to the count of dropping patterns of 2^n+c(2^n), with the c(2^n) as mentioned with A177789. The positions of the zeros of that variant sequence might be a close variant of A054414, again with a(0)=0 (not properly checked yet). - Ruud H.G. van Tol, Nov 28 2021
It appears that the proportion of zeros is 1-log(2)/log(3) = 36.907...%. - Jesse Randall, Oct 10 2024

Examples

			a(1) = 0 since there exists no odd number whose msa is 1;
a(2) = 1 since there is only one odd number, 5 with k=2 2k+1, with k less than 2^2 whose msa is 2;
a(3) = 0 since there exists no odd number whose msa is 3;
a(4) = 1 since there is only one number, 1, less than 2^(4+1) whose msa is 4;
a(5) = 2 since there are two numbers, 11 & 23, less than 2^(4+1) whose msa is 4; etc.

Crossrefs

Cf. A260590, A054414, A100982, A186009, A186008, A177789, A346640.

Programs

  • Mathematica
    msa[n_] := If[ OddQ@ n, (3n + 1)/2, n/2]; f[n_] := Block[{k = 2n + 1}, Length@ NestWhileList[ msa@# &, k, # >= k &] - 1]; g[n_] := Length@ Select[ Range[ 2^(n - 1)], f@# == n &]; Array[ g, 20]

Extensions

a(31) onwards from Jesse Randall, Sep 09 2024

A260592 a(n) = binary odd/even encoding of the iterates in the modified Syracuse algorithm (msa) starting with 2n+1 and continuing up to (but not including) the first iterate less than 2n+1.

Original entry on oeis.org

1100, 10, 1110100, 10, 11010, 10, 1111000, 10, 1100, 10, 11100, 10, 11011111010110111011110100111011011111100111100010101000100, 10, 11111010110111011110100111011011111100111100010101000100, 10, 1100, 10, 11101100, 10, 11010, 10
Offset: 1

Views

Author

Joseph K. Horn and Robert G. Wilson v, Jul 31 2015

Keywords

Comments

For the msa mapping see A260590; if x is odd append 1 and if x is even append 0.
The binary length of a(n) is A260590(n).
For even numbers, 2n, append to f(n) a 0. Example: f(10) = 0, f(5) = 010.
Tallying all the ones and zeros, there appear to be five ones for every four zeros.
Terms sorted in increasing order and duplicates removed: 10, 1100, 11010, 11100, 1101100, 1110100, 1111000, ...
Since msa always starts with an odd number every binary encoding starts with digit 1 and has at least two digits. - Hartmut F. W. Hoft, Nov 05 2015

Examples

			a(1) = 1100 since A260590(1) is 4, the four operations are, in order following the msa mapping scheme: (3x+1)/2, (3x+1)/2, x/2, and finishing with a x/2 mapping.

Crossrefs

Cf. A005408, A176999, A260590.

Programs

  • Mathematica
    f[n_] := Block[{k = 2n + 1, lst = {}}, While[k > 2n, If[ OddQ@ k, k = (3k + 1)/2; AppendTo[ lst, 1], k /= 2; AppendTo[ lst, 0]]]; FromDigits@ lst]; Array[f, 22]

Formula

a(n) = b_1 b_2 ... b_k, the binary k-digit number where b_j = 1 when the j-th iterate of msa is odd and b_j = 0 when it is even, where the first k iterates exceed 2n+1, but the (k+1)-st iterate is less than 2n+1. - Hartmut F. W. Hoft, Nov 05 2015

Extensions

Name change by Hartmut F. W. Hoft, Nov 05 2015
Showing 1-4 of 4 results.

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