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 1-8 of 8 results.

A285488 Inverse permutation to A285487.

Original entry on oeis.org

1, 3, 165, 5, 31, 7, 25, 643, 167, 33, 23, 9, 27, 17, 59, 645, 29, 11, 151, 35, 53, 13, 153, 169, 161, 15, 595, 19, 155, 37, 157, 647, 51, 129, 42, 171, 159, 21, 55, 135, 163, 45, 213, 131, 61, 137, 215, 173, 217, 139, 57, 133, 219, 175, 44, 143, 63, 141, 221
Offset: 1

Views

Author

Rémy Sigrist, Apr 19 2017

Keywords

Examples

			A285487(1) = 1,    hence a(1)    = 1.
A285487(2) = 2310, hence a(2310) = 2.
A285487(3) = 2,    hence a(2)    = 3.
A285487(4) = 1155, hence a(1155) = 4.
A285487(5) = 4,    hence a(4)    = 5.
A285487(6) = 1365, hence a(1365) = 6.
A285487(7) = 6,    hence a(6)    = 7.
A285487(8) = 385,  hence a(385)  = 8.
A285487(9) = 12,   hence a(12)   = 9.

Links

Crossrefs

Cf. A285487.

A285655 Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms has at least 6 distinct prime factors.

Original entry on oeis.org

1, 30030, 2, 15015, 4, 19635, 6, 5005, 12, 6545, 18, 7315, 24, 7735, 22, 1365, 34, 1155, 26, 1785, 38, 2145, 14, 2805, 28, 3135, 42, 715, 84, 935, 78, 385, 102, 455, 66, 595, 114, 770, 39, 1190, 33, 910, 51, 1330, 69, 1430, 21, 1870, 57, 1540, 87, 1610, 93
Offset: 1

Views

Author

Rémy Sigrist, Apr 23 2017

Keywords

Comments

This sequence can always be extended with a multiple of 30030 = 2*3*5*7*11*13; after a term that has at least 6 distinct prime factors, we can extend the sequence with the least unused number; as there are infinitely many numbers with at least 6 distinct prime factors, this sequence is a permutation of the natural numbers (with inverse A285656).
Conjecturally, a(n) ~ n.
The first fixed points are: 1, 39, 1344, 1350, 3556, 3560, 5738, 6974, 15668585, 15668673, 15668787.
For any k>0, let d_k be the lexicographically earliest sequence of distinct terms such that the product of two consecutive terms has at least k distinct prime factors; in particular we have:
- d_1 = A000027 (the natural numbers),
- d_5 = A285487,
- d_6 = a (this sequence).
For any k>0:
- d_k is a permutation of the natural numbers,
- d_k(1) = 1 and d_k(2) = A002110(k),
- conjecturally: d_k(n) ~ n.

Examples

			The first terms, alongside the primes p dividing a(n)*a(n+1), are:
n       a(n)    p
--      ----    ------------------
1       1       2, 3, 5, 7, 11, 13
2       30030   2, 3, 5, 7, 11, 13
3       2       2, 3, 5, 7, 11, 13
4       15015   2, 3, 5, 7, 11, 13
5       4       2, 3, 5, 7, 11,     17
6       19635   2, 3, 5, 7, 11,     17
7       6       2, 3, 5, 7, 11, 13
8       5005    2, 3, 5, 7, 11, 13
9       12      2, 3, 5, 7, 11,     17
10      6545    2, 3, 5, 7, 11,     17
11      18      2, 3, 5, 7, 11,         19
12      7315    2, 3, 5, 7, 11,         19
13      24      2, 3, 5, 7,     13, 17
14      7735    2,    5, 7, 11, 13, 17
15      22      2, 3, 5, 7, 11, 13
16      1365    2, 3, 5, 7,     13, 17
17      34      2, 3, 5, 7, 11,     17
18      1155    2, 3, 5, 7, 11, 13
19      26      2, 3, 5, 7,     13, 17
20      1785    2, 3, 5, 7,         17, 19

Links

Crossrefs

Cf. A000027, A002110, A285487, A285656 (inverse).

A288164 Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, a(n)*a(n+2) has at least 5 distinct prime factors.

Original entry on oeis.org

1, 2, 2310, 1155, 3, 4, 770, 1365, 6, 8, 385, 1785, 12, 10, 455, 231, 18, 20, 595, 273, 22, 30, 105, 77, 26, 60, 165, 91, 14, 66, 195, 35, 28, 78, 255, 55, 38, 42, 210, 65, 11, 84, 390, 85, 7, 114, 330, 70, 13, 33, 420, 130, 17, 21, 462, 110, 5, 39, 546, 140
Offset: 1

Views

Author

Rémy Sigrist, Jun 16 2017

Keywords

Comments

This sequence is a permutation of the natural numbers, with inverse A288799.
Conjecturally, a(n) ~ n.
For k >= 0, let f_k be the lexicographically earliest sequence of distinct positive terms such that, for any n > 0, a(n)*a(n+k) has at least 5 distinct prime factors.
In particular, we have:
- f_0 = the numbers with at least 5 distinct prime factors,
- f_1 = A285487,
- f_2 = a (this sequence),
- f_3 = A288171.
If k > 0, then:
- f_k is a permutation of the natural numbers,
- f_k(i) = i for any i <= k,
- f_k(k+1) = A002110(5),
- conjecturally, f_k(n) ~ n.

Examples

			The first terms, alongside the primes p dividing a(n)*a(n+2), are:
n       a(n)    p
--      ----    --------------
1       1       2, 3, 5, 7, 11
2       2       2, 3, 5, 7, 11
3       2310    2, 3, 5, 7, 11
4       1155    2, 3, 5, 7, 11
5       3       2, 3, 5, 7, 11
6       4       2, 3, 5, 7,     13
7       770     2, 3, 5, 7, 11
8       1365    2, 3, 5, 7,     13
9       6       2, 3, 5, 7, 11
10      8       2, 3, 5, 7,         17
11      385     2, 3, 5, 7, 11
12      1785    2, 3, 5, 7,         17
13      12      2, 3, 5, 7,     13
14      10      2, 3, 5, 7, 11
15      455     2, 3, 5, 7,     13
16      231     2, 3, 5, 7, 11
17      18      2, 3, 5, 7,         17
18      20      2, 3, 5, 7,     13
19      595     2,    5, 7, 11,     17
20      273     2, 3, 5, 7,     13
21      22      2, 3, 5, 7, 11
22      30      2, 3, 5, 7, 11
23      105     2, 3, 5, 7,     13

Links

Crossrefs

Cf. A002110, A285487, A288171, A288799 (inverse).

A288923 Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms has at least 6 prime factors (counted with multiplicity).

Original entry on oeis.org

1, 64, 2, 32, 3, 48, 4, 16, 6, 24, 8, 12, 18, 20, 27, 28, 30, 36, 9, 40, 10, 54, 14, 56, 15, 60, 21, 72, 5, 80, 7, 96, 11, 108, 13, 112, 17, 120, 19, 128, 22, 81, 25, 84, 26, 88, 33, 90, 34, 100, 35, 104, 38, 126, 39, 132, 42, 44, 45, 50, 52, 63, 66, 68, 70
Offset: 1

Views

Author

Rémy Sigrist, Jun 19 2017

Keywords

Comments

The number of prime factors counted with multiplicity is given by A001222.
This sequence is a permutation of the natural numbers, with inverse A288924.
Conjecturally, a(n) ~ n.
For a function g over the natural numbers and a constant K, let f(g,K) be the lexicographically earliest sequence of distinct positive terms such that, for any n > 0, g( f(g,K)(n) * f(g,K)(n+1) ) >= K. In particular we have:
- f(bigomega, 6) = a (this sequence), where bigomega = A001222,
- f(tau, 34) = A288921, where tau = A000005,
- f(omega, 5) = A285487, where omega = A001221,
- f(omega, 6) = A285655, where omega = A001221.
Some of these sequences have similar graphical features.

Examples

			The first terms, alongside a(n) * a(n+1) and its number of prime divisors counted with multiplicity, are:
   n   a(n)   a(n)*a(n+1)   Bigomega
  --   ----   -----------   --------
   1     1         64           6
   2    64        128           7
   3     2         64           6
   4    32         96           6
   5     3        144           6
   6    48        192           7
   7     4         64           6
   8    16         96           6
   9     6        144           6
  10    24        192           7
  11     8         96           6
  12    12        216           6
  13    18        360           6
  14    20        540           6
  15    27        756           6
  16    28        840           6
  17    30       1080           7
  18    36        324           6
  19     9        360           6
  20    40        400           6

Links

Crossrefs

Cf. A001221, A001222, A285487, A285655, A288921, A288924 (inverse).

A280659 Lexicographically earliest sequence of distinct positive terms such that the sum of two consecutive terms has at least 5 distinct prime factors.

Original entry on oeis.org

1, 2309, 421, 1889, 841, 1469, 1261, 1049, 1681, 629, 2101, 209, 2521, 1769, 541, 2189, 121, 2609, 961, 1349, 1381, 929, 1801, 509, 2221, 89, 2641, 1649, 661, 2069, 241, 2489, 1081, 1229, 1501, 809, 1921, 389, 2341, 1949, 361, 2369, 1201, 1109, 1621, 689, 2041
Offset: 1

Views

Author

Rémy Sigrist, Apr 25 2017

Keywords

Comments

Conjecturally: this sequence is a permutation of the natural numbers, and a(n) ~ n.
The first fixed points are: 1, 7379, 7730, 7765, 7846, 9535, 9903, 11604, 11631, 11741, 12674, 15549, 15824, 16670, 16745, 16800, 16806, 16841.
This sequence has similarities with A285487: here we consider the sum of consecutive terms, there the product of consecutive terms.
From Rémy Sigrist, Jul 16 2017: (Start)
The scatterplot of the first terms presents rectangular clusters of points near the origin; these clusters seem to correspond to indexes n satisfying a(n) + a(n+1) < 2 * prime#(5) (where prime(k)# = A002110(k)).
Near the origin, we also have ranges of more than hundred consecutive terms where the function b satisfying b(n) = lpf(a(n)) (where lpf = A020639) is constant (and equals 2, 3 or 5).
These features are highlighted in the alternate scatterplots provided in the Links section.
There features are also visible in the scatterplots of variants of this sequence where we increase the minimum number of distinct prime factors required for the sum of two consecutive terms.
(End)

Examples

			The first terms, alongside the primes p dividing a(n)+a(n+1), are:
n       a(n)    p
--      ----    --------------
1       1       2, 3, 5, 7, 11
2       2309    2, 3, 5, 7,     13
3       421     2, 3, 5, 7, 11
4       1889    2, 3, 5, 7,     13
5       841     2, 3, 5, 7, 11
6       1469    2, 3, 5, 7,     13
7       1261    2, 3, 5, 7, 11
8       1049    2, 3, 5, 7,     13
9       1681    2, 3, 5, 7, 11
10      629     2, 3, 5, 7,     13
11      2101    2, 3, 5, 7, 11
12      209     2, 3, 5, 7,     13
13      2521    2, 3, 5,    11, 13
14      1769    2, 3, 5, 7, 11
15      541     2, 3, 5, 7,     13
16      2189    2, 3, 5, 7, 11
17      121     2, 3, 5, 7,     13
18      2609    2, 3, 5, 7,         17
19      961     2, 3, 5, 7, 11
20      1349    2, 3, 5, 7,     13

Links

Crossrefs

Cf. A002110, A020639, A285487.

A285744 Lexicographically earliest sequence of distinct positive terms such that, for any n>0, n*a(n) has at least 5 distinct prime factors.

Original entry on oeis.org

2310, 1155, 770, 1365, 462, 385, 330, 1785, 910, 231, 210, 455, 420, 165, 154, 1995, 390, 595, 510, 273, 110, 105, 546, 665, 714, 255, 1190, 195, 570, 77, 630, 2145, 70, 285, 66, 715, 660, 315, 140, 357, 690, 55, 780, 345, 182, 399, 798, 805, 858, 429, 130
Offset: 1

Views

Author

Rémy Sigrist, Apr 25 2017

Keywords

Comments

If n has at least 5 distinct prime factors, then a(n) is the least unused number; as there are infinitely many numbers with at least 5 distinct prime factors, this sequence is a permutation of the natural numbers.
The inverse of this sequence is the sequence itself.
The first fixed points are: 40755, 42966, 54285, 54740, 55965, 56070, 66045, 66066, 70035, 70350, 73815, 73920 (note that the fixed points have at least 5 distinct prime factors).
Conjecturally, a(n) ~ n.
This sequence has similarities with A285487: here n*a(n) has at least 5 distinct prime factors, there a(n)*a(n+1) has at least 5 distinct prime factors.

Examples

			The first terms, alongside the primes p dividing n*a(n), are:
n       a(n)    p
--      ----    --------------
1       2310    2, 3, 5, 7, 11
2       1155    2, 3, 5, 7, 11
3       770     2, 3, 5, 7, 11
4       1365    2, 3, 5, 7,     13
5       462     2, 3, 5, 7, 11
6       385     2, 3, 5, 7, 11
7       330     2, 3, 5, 7, 11
8       1785    2, 3, 5, 7,         17
9       910     2, 3, 5, 7,     13
10      231     2, 3, 5, 7, 11
11      210     2, 3, 5, 7, 11
12      455     2, 3, 5, 7,     13
13      420     2, 3, 5, 7,     13
14      165     2, 3, 5, 7, 11
15      154     2, 3, 5, 7, 11
16      1995    2, 3, 5, 7,             19
17      390     2, 3, 5,        13, 17
18      595     2, 3, 5, 7,         17
19      510     2, 3, 5,            17, 19
20      273     2, 3, 5, 7,     13

Links

Crossrefs

Cf. A285487.

A306854 Lexicographically earliest sequence of distinct positive terms such that the product of two consecutive terms has at least 5 distinct Fermi-Dirac prime factors.

Original entry on oeis.org

1, 840, 3, 280, 9, 120, 7, 216, 5, 168, 11, 210, 4, 270, 13, 264, 15, 56, 27, 40, 21, 72, 33, 70, 12, 90, 28, 30, 36, 42, 20, 54, 35, 24, 45, 66, 52, 96, 44, 78, 55, 84, 10, 108, 14, 60, 18, 105, 8, 135, 22, 140, 6, 180, 26, 132, 32, 156, 34, 165, 38, 189, 46
Offset: 1

Views

Author

Rémy Sigrist, Mar 13 2019

Keywords

Comments

This sequence is a variant of A285487. Both sequences are permutations of the natural numbers and have similar graphical features.

Examples

			The first terms, alongside the Fermi-Dirac factorization of a(n) * a(n+1), are:
  n   a(n)  a(n) * a(n+1)
  --  ----  -------------
   1     1  2^(2^0) * 2^(2^1) * 3^(2^0) * 5^(2^0)  * 7^(2^0)
   2   840  2^(2^0) * 2^(2^1) * 3^(2^1) * 5^(2^0)  * 7^(2^0)
   3     3  2^(2^0) * 2^(2^1) * 3^(2^0) * 5^(2^0)  * 7^(2^0)
   4   280  2^(2^0) * 2^(2^1) * 3^(2^1) * 5^(2^0)  * 7^(2^0)
   5     9  2^(2^0) * 2^(2^1) * 3^(2^0) * 3^(2^1)  * 5^(2^0)
   6   120  2^(2^0) * 2^(2^1) * 3^(2^0) * 5^(2^0)  * 7^(2^0)
   7     7  2^(2^0) * 2^(2^1) * 3^(2^0) * 3^(2^1)  * 7^(2^0)
   8   216  2^(2^0) * 2^(2^1) * 3^(2^0) * 3^(2^1)  * 5^(2^0)
   9     5  2^(2^0) * 2^(2^1) * 3^(2^0) * 5^(2^0)  * 7^(2^0)
  10   168  2^(2^0) * 2^(2^1) * 3^(2^0) * 7^(2^0)  * 11^(2^0)
  11    11  2^(2^0) * 3^(2^0) * 5^(2^0) * 7^(2^0)  * 11^(2^0)
  12   210  2^(2^0) * 2^(2^1) * 3^(2^0) * 5^(2^0)  * 7^(2^0)

Links

Crossrefs

Cf. A064547, A285487, A306856 (inverse).

Programs

  • PARI
    See Links section.

Formula

A064547(a(n) * a(n+1)) >= 5.

A306864 Lexicographically earliest sequence of distinct positive terms such that among the prime divisors of the product of two consecutive terms there are at least 4 runs of consecutive prime numbers.

Original entry on oeis.org

1, 1870, 2, 935, 4, 1045, 8, 1235, 10, 187, 20, 209, 40, 247, 14, 299, 21, 377, 28, 391, 22, 85, 44, 95, 26, 115, 34, 55, 38, 65, 46, 91, 57, 182, 19, 110, 17, 220, 23, 130, 29, 170, 11, 190, 13, 230, 31, 238, 37, 260, 41, 266, 39, 133, 52, 145, 68, 155, 76
Offset: 1

Views

Author

Rémy Sigrist, Mar 14 2019

Keywords

Comments

This sequence is a variant of A285487.
This sequence is likely a permutation of the natural numbers.

Examples

			The first terms, alongside the corresponding runs, are:
   n   a(n)  runs in a(n)*a(n+1)
  ---  ----  -------------------
    1     1  2, 5, 11, 17
    2  1870  2, 5, 11, 17
    3     2  2, 5, 11, 17
    4   935  2, 5, 11, 17
    5     4  2, 5, 11, 19
    6  1045  2, 5, 11, 19
    7     8  2, 5, 13, 19
    8  1235  2, 5, 13, 19
    9    10  2, 5, 11, 17
   10   187  2, 5, 11, 17
   11    20  2, 5, 11, 19
   12   209  2, 5, 11, 19
  ...
   32    91  3, 7, 13, 19
   33    57  2-3, 7, 13, 19
   34   182  2, 7, 13, 19
  ...
  662  1222  2, 7, 13, 47
  663   448  2, 7, 17, 73
  664  1241  2-3, 7-11, 17, 73
  665   462  2-3, 7-11, 29, 43
  666  1247  3, 17, 29, 43
  ...

Links

Crossrefs

Cf. A287170, A285487.

Programs

  • PARI
    See Links section.

Formula

A287170(a(n) * a(n+1)) >= 4.
Showing 1-8 of 8 results.

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