A338349 -id:A338349 - cp's OEIS frontend

A338338 Lexicographically earliest infinite sequence of distinct positive numbers such that for any prime p, any run of consecutive multiples of p has length exactly 3.

1, 2, 4, 6, 3, 9, 5, 10, 20, 8, 7, 14, 28, 12, 15, 30, 40, 16, 11, 22, 44, 18, 21, 42, 56, 24, 27, 33, 55, 110, 50, 26, 13, 39, 36, 48, 32, 17, 34, 68, 38, 19, 57, 45, 60, 70, 84, 63, 51, 85, 170, 80, 46, 23, 69, 54, 66, 88, 77, 35, 105, 75, 72, 52, 78, 117

Offset: 1

N. J. A. Sloane, Oct 27 2020

nonn

If a prime p divides a(n), then there is a run of exactly three terms (one of which is a(n)) that are divisible by p.
If "three" is changed to "two", we get A280864.
Conjecture: This is a permutation of the positive integers.

			After 1,2,4,6,3, we have had two successive multiples of 3, so the next term must be a multiple of 3 we have not yet seen, hence 9. The following term is then the smallest number not yet seen which is not a multiple of 3, hence 5.

Rémy Sigrist, Table of n, a(n) for n = 1..20000
Rémy Sigrist, PARI program for A338338

A338339-A338349, A338440, A338449, A338450, and A338451 analyze this sequence from various points of view.

Cf. A280864, A338441.

PARI
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See Links section.
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Corrected and extended by Rémy Sigrist, Oct 27 2020

A338339 Primes p in A338338 that occur at the start of a run of three terms in A338338 that are divisible by p.

Original entry on oeis.org

2, 5, 7, 11, 17, 31, 37, 41, 53, 67, 71, 79, 89, 101, 109, 131, 151, 173, 181, 197, 199, 211, 241, 257, 263, 277, 281, 293, 311, 347, 359, 383, 421, 431, 443, 487, 521, 523, 593, 607, 643, 659, 701, 709, 739, 809, 811, 827, 883, 971, 983, 1009, 1019, 1031, 1039, 1061, 1069, 1109

Offset: 1

N. J. A. Sloane, Oct 27 2020

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C program for A338339

Cf. A338338-A338349.

C
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See Links section.
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A338340 Numbers k such that A338338(k) is a prime p that starts a run of three terms in A338338 that are divisible by p.

Original entry on oeis.org

2, 7, 11, 19, 38, 78, 82, 91, 125, 157, 161, 186, 206, 260, 290, 331, 377, 430, 453, 493, 502, 514, 601, 610, 619, 675, 684, 711, 744, 797, 822, 946, 1039, 1061, 1087, 1215, 1313, 1317, 1494, 1557, 1736, 1765, 1930, 1939, 2053, 2217, 2231, 2260, 2522, 2710, 2719, 2846, 2904

Offset: 1

N. J. A. Sloane, Oct 27 2020

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A338340

Cf. A338338-A338349.

PARI
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See Links section.
```

A338342 Numbers k such that A338338(k) is a prime p in the middle of a run of three terms in A338338 that are divisible by p.

Original entry on oeis.org

5, 33, 42, 54, 73, 95, 120, 145, 152, 175, 190, 236, 264, 270, 294, 325, 335, 341, 372, 381, 387, 448, 474, 480, 552, 562, 568, 586, 605, 649, 655, 688, 739, 748, 768, 790, 811, 817, 921, 934, 973, 979, 1001, 1027, 1033, 1065, 1071, 1106, 1135, 1152, 1168, 1200, 1219, 1247, 1264

Offset: 1

N. J. A. Sloane, Oct 27 2020

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A338342

Cf. A338338-A338349.

PARI
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See Links section.
```

A338344 Numbers k such that A338338(k) is a prime p that ends a run of three terms in A338338 that are divisible by p.

Original entry on oeis.org

429, 600, 796, 945, 1214, 1556, 3376, 3828, 4282, 4867, 5406, 6209, 6280, 6741, 6901, 7281, 7589, 7642, 7982, 8361, 8491, 8522, 8636, 8825, 8904, 9060, 9109, 9550, 9642, 9784, 10033, 10102, 10310, 11405, 11526, 11990, 12140, 12210, 12254, 12392, 12568, 13499, 13550

Offset: 1

N. J. A. Sloane, Oct 27 2020

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C program for A338344

Cf. A338338-A338349.

C
```
See Links section.
```

A338348 Records in A338338.

Original entry on oeis.org

1, 2, 4, 6, 9, 10, 20, 28, 30, 40, 44, 56, 110, 170, 190, 248, 370, 424, 430, 470, 670, 710, 890, 1310, 1336, 1384, 1448, 1590, 1592, 2110, 2630, 2770, 2810, 3590, 3790, 3832, 3896, 4168, 4184, 4420, 5126, 6010, 7390, 8110, 9710, 10090, 10186, 12310, 13190, 15710, 16210

Offset: 1

N. J. A. Sloane, Oct 27 2020

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C program for A338348

Cf. A338338-A338349.

C
```
See Links section.
```

A338341 Primes p in A338338 that occur in the middle of a run of three terms in A338338 that are divisible by p.

Original entry on oeis.org

3, 13, 19, 23, 29, 43, 47, 59, 61, 73, 83, 97, 103, 107, 113, 127, 137, 139, 149, 157, 163, 179, 191, 193, 223, 227, 229, 233, 251, 269, 271, 283, 307, 313, 317, 331, 349, 353, 367, 373, 389, 397, 401, 409, 419, 433, 439, 449, 457, 461, 463, 467, 491, 499, 503, 509, 541, 547, 557

Offset: 1

N. J. A. Sloane, Oct 27 2020

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A338341

Cf. A338338-A338349.

PARI
```
See Links section.
```

A338343 Primes p in A338338 that occur at the end of a run of three terms in A338338 that are divisible by p.

Original entry on oeis.org

167, 239, 337, 379, 479, 601, 1181, 1301, 1511, 1721, 1871, 2069, 2099, 2239, 2281, 2377, 2467, 2477, 2609, 2749, 2803, 2843, 2879, 2917, 2963, 3049, 3089, 3209, 3221, 3299, 3361, 3389, 3499, 3907, 3923, 4049, 4127, 4153, 4201, 4271, 4327, 4637, 4651, 4673, 4733, 4871

Offset: 1

N. J. A. Sloane, Oct 27 2020

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, C program for A338343

Cf. A338338-A338349.

C
```
See Links section.
```

A338345 Numbers k such that A338338(k) is a prime.

Original entry on oeis.org

2, 5, 7, 11, 19, 33, 38, 42, 54, 73, 78, 82, 91, 95, 120, 125, 145, 152, 157, 161, 175, 186, 190, 206, 236, 260, 264, 270, 290, 294, 325, 331, 335, 341, 372, 377, 381, 387, 429, 430, 448, 453, 474, 480, 493, 502, 514, 552, 562, 568, 586, 600, 601, 605, 610, 619, 649, 655, 675

Offset: 1

N. J. A. Sloane, Oct 27 2020

nonn

This is the sorted union of A338340, A338342, and A338344.

Note that the primes in A338338 appear in their natural order.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A338345

Cf. A338338-A338349.

PARI
```
See Links section.
```

A338346 1, followed by first differences of A338340.

Original entry on oeis.org

1, 5, 4, 8, 19, 40, 4, 9, 34, 32, 4, 25, 20, 54, 30, 41, 46, 53, 23, 40, 9, 12, 87, 9, 9, 56, 9, 27, 33, 53, 25, 124, 93, 22, 26, 128, 98, 4, 177, 63, 179, 29, 165, 9, 114, 164, 14, 29, 262, 188, 9, 127, 58, 10, 10, 58, 30, 97, 268, 229, 223, 28, 9, 97, 62, 9, 67, 87, 43, 52

Offset: 1

N. J. A. Sloane, Oct 27 2020

nonn

A338338 has a natural division into blocks, where the blocks (after the initial 1) start at the primes described in A338339 and A338340. The first few blocks are
[1], [2, 4, 6, 3, 9], [5, 10, 20, 8], [7, 14, 28, 12, 15, 30, 40, 16], [11, 22, 44, 18, ..., 36, 48, 32], [17, ...
and the present sequence gives the lengths of the successive blocks.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A338338-A338349.

cp's OEIS Frontend

A338338 Lexicographically earliest infinite sequence of distinct positive numbers such that for any prime p, any run of consecutive multiples of p has length exactly 3.

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A338339 Primes p in A338338 that occur at the start of a run of three terms in A338338 that are divisible by p.

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C

A338340 Numbers k such that A338338(k) is a prime p that starts a run of three terms in A338338 that are divisible by p.

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A338342 Numbers k such that A338338(k) is a prime p in the middle of a run of three terms in A338338 that are divisible by p.

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A338344 Numbers k such that A338338(k) is a prime p that ends a run of three terms in A338338 that are divisible by p.

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C

A338348 Records in A338338.

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C

A338341 Primes p in A338338 that occur in the middle of a run of three terms in A338338 that are divisible by p.

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PARI

A338343 Primes p in A338338 that occur at the end of a run of three terms in A338338 that are divisible by p.

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C

A338345 Numbers k such that A338338(k) is a prime.

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A338346 1, followed by first differences of A338340.

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