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.

A261255 Where MU-numbers (cf. A007335) occur in A003586 (3-smooth numbers).

Original entry on oeis.org

2, 3, 5, 8, 10, 11, 15, 16, 20, 24, 25, 26, 31, 33, 38, 39, 44, 45, 47, 53, 54, 57, 61, 64, 70, 71, 72, 75, 80, 83, 87, 90, 92, 96, 101, 104, 105, 109, 113, 115, 119, 123, 125, 129, 134, 138, 140, 144, 145, 149, 151, 156, 161, 165, 166, 168, 173, 178, 180
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 13 2015

Keywords

Examples

			.   n |   A007335(n)     | a(n) | A003586(a(n))
.  ---+------+-----------+------+--------------
.   1 |    2 |         2 |    2 |            2
.   2 |    3 |         3 |    3 |            3
.   3 |    6 |     2 * 3 |    5 |            6
.   4 |   12 |   2^2 * 3 |    8 |           12
.   5 |   18 |   2 * 3^2 |   10 |           18
.   6 |   24 |   2^3 * 3 |   11 |           24
.   7 |   48 |   2^4 * 3 |   15 |           48
.   8 |   54 |   2 * 3^3 |   16 |           54
.   9 |   96 |   2^5 * 3 |   20 |           96
.  10 |  162 |   2 * 3^4 |   24 |          162
.  11 |  192 |   2^6 * 3 |   25 |          192
.  12 |  216 | 2^3 * 3^3 |   26 |          216 .

Links

Crossrefs

Cf. A007335, A003586.

Programs

  • Haskell
    import Data.List (findIndex); import Data.Maybe (fromJust)
a261255 n = fromJust (findIndex (== a007335 n) a003586_list) + 1

Formula

A007335(n) = A003586(a(n)).

A054540 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the six simple ratios of musical harmony: 6/5, 5/4, 4/3, 3/2, 8/5 and 5/3.

Original entry on oeis.org

1, 2, 3, 5, 7, 12, 19, 31, 34, 53, 118, 171, 289, 323, 441, 612, 730, 1171, 1783, 2513, 4296, 12276, 16572, 20868, 25164, 46032, 48545, 52841, 73709, 78005, 151714, 229719, 537443, 714321, 792326, 944040, 1022045, 1251764, 3755292, 3985011
Offset: 0

Views

Author

Mark William Rankin (MarkRankin95511(AT)Yahoo.com), Apr 09 2000; Dec 17 2000

Keywords

Comments

The sequence was found by a computer search of all of the equal divisions of the octave from 1 to over 3985011. There seems to be a hidden aspect or mystery here: what is it about the more and more harmonious equal temperaments that causes them to express themselves collectively as a perfect, self-accumulating recurrent sequence?
From Eliora Ben-Gurion, Dec 15 2022: (Start)
The answer is because temperament mappings can be added. If harmonic correspondences are written in a bra, that is
Example: a tuning with 118 equal steps to the octave has a second harmonic on the 118th step by definition, the third harmonic is approximated with 187 steps, and the fifth is with 274 steps, which leads to <118 187 274]. A 171 equal division system will have a corresponding bra <171 271 397]. When these two are added, we obtain <289 458 671], which is exactly how the 2nd, 3rd, and 5th harmonics are represented in 289 equal divisions of the octave. (End)

Examples

			34 = 31 + the earlier term 3. Again, 118 = 53 + the earlier terms 34 and 31.

Links

Crossrefs

Cf. A001149, A018065, A001856, A002858, A007335, A060525, A060526, A060527.

Formula

Stochastic recurrence rule - the next term equals the current term plus one or more previous terms: a(n+1) = a(n) + a(n-x) + ... + a(n-y) + ... + a(n-z), etc.

A000423 a(n) is smallest number > a(n-1) of form a(i)*a(j), i < j < n.

Original entry on oeis.org

2, 3, 6, 12, 18, 24, 36, 48, 54, 72, 96, 108, 144, 162, 192, 216, 288, 324, 384, 432, 486, 576, 648, 768, 864, 972, 1152, 1296, 1458, 1536, 1728, 1944, 2304, 2592, 2916, 3072, 3456, 3888, 4374, 4608, 5184, 5832, 6144, 6912, 7776, 8748, 9216, 10368, 11664
Offset: 1

Views

Author

R. Muller

Keywords

Comments

Sequence contains 2, 3 and all numbers of form 2^a*3^b where a >= 1 and b >= 1. - David W. Wilson, Aug 15 1996
Main entry for this sequence is A033845, which is this sequence starting at 6. - Charles R Greathouse IV, Feb 27 2012

References

  • Amarnath Murthy, The sum of the reciprocals of the Smarandache multiplicative sequence, (to be published in Smarandache Notions Journal).
  • F. Smarandache, "Properties of the Numbers", University of Craiova Archives, 1975; Arizona State University Special Collections, Tempe, AZ
  • M. Myers, Smarandache Multiplicative Numbers, in Memorables 1998, Bristol Banner Books, Bristol, p. 37, 1998.

Links

Crossrefs

Subsequence of A003586 (3-smooth numbers).
A007335 and A033845 are subsequences.

Programs

  • Mathematica
    a[1] = 2; a[2] = 3; a[n_] := a[n] = For[k = a[n - 1] + 1, True, k++, If[ AnyTrue[Table[a[i] a[j], {i, 1, n-2}, {j, i+1, n-1}] // Flatten, # == k& ], Return[k]]]; Table[an = a[n]; Print[an]; an, {n, 1, 50}] (* Jean-François Alcover, Feb 08 2016 *)

Formula

Sum_{n>=1} 1/a(n) = 4/3. - Amiram Eldar, Jul 31 2022

Extensions

More terms from David W. Wilson, Aug 15 1996

A060301 For n>3: a(n) is a multiple of three distinct earlier terms.

Original entry on oeis.org

2, 3, 5, 30, 180, 300, 450, 1080, 1800, 2700, 3000, 4500, 6480, 6750, 10800, 16200, 18000, 27000, 30000, 38880, 40500, 45000, 64800, 67500, 97200, 101250, 108000, 162000, 180000, 233280, 243000, 270000, 300000, 388800, 405000, 450000, 583200
Offset: 1

Views

Author

Henry Bottomley, Mar 28 2001

Keywords

Examples

			a(15) = 10800 = 2*30*180 = a(1)*a(4)*a(5).

Links

Crossrefs

Cf. A000423.

Formula

a(n) is of the form 2^b*3^c*5^d where either b+c+d=1, or where each of b, c and d are positive and the largest of the three is an odd number less than the sum of the other two. For n>3: a(n) is 30 times a power of 6 times a power of 10 times a power of 15.
Showing 1-4 of 4 results.

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