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 1-2 of 2 results.

A186107 Numerator of the frequency of the n-th dropping time in the Collatz iteration.

Original entry on oeis.org

1, 1, 1, 1, 3, 7, 3, 15, 85, 173, 119, 961, 663, 8045, 17637, 51033, 54475, 312455, 663535, 950235, 5936673, 1684037, 39993895, 87986917, 128989251, 205059181, 949737339, 2861515293, 400296173, 19018424205
Offset: 1

Views

Author

T. D. Noe, Feb 12 2011

Keywords

Comments

The possible dropping times are in A020914. The denominators are in A186108. The cumulative frequency of the dropping time is A186109(n)/A186110(n).

Examples

			The frequencies are 1/2, 1/4, 1/16, 1/16, 3/128, 7/256, 3/256, 15/2048, 85/8192,....

Links

Crossrefs

Cf. A126241 (dropping times)

Formula

a(n) = numerator of A186009(n) / 2^A020914(n-1).

A186109 Numerator of the cumulative frequency of the dropping time in the Collatz iteration.

Original entry on oeis.org

1, 3, 13, 7, 115, 237, 15, 1935, 7825, 31473, 31711, 254649, 15957, 2050541, 8219801, 16490635, 33035745, 132455435, 530485275, 1061920785, 4253619813, 4256987887, 34095896991, 136471574881, 273072139013, 136638599097, 2187167322891, 4377196161075, 4378797345767, 35049397190341
Offset: 1

Views

Author

T. D. Noe, Feb 12 2011

Keywords

Comments

The possible dropping times are in A020914. The denominators are in A186110. The frequency of the n-th dropping time is A186107(n)/A186108(n).
Riho Terras' classic paper about the Collatz problem shows the decimal values of 2(1-c(k)) in Table A, where c(k) is the cumulative frequency of dropping times <= k.

Examples

			The cumulative frequencies are 1/2, 3/4, 13/16, 7/8, 115/128, 237/256, 15/16, 1935/2048, 7825/8192, ... .

Links

Crossrefs

Cf. A126241 (dropping times).

Formula

a(n) = numerator of Sum_{k=1..n} A186009(k) / 2^A020914(k-1).
Showing 1-2 of 2 results.

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

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