A342068 -id:A342068 - cp's OEIS frontend

A342069 a(n) is the smallest k such that A342068(k) = n.

1, 5, 12, 4, 2, 6, 31, 28, 3, 13, 169, 729, 3128, 2245, 4660, 2524, 5198, 519, 40016, 48916, 12350, 45362, 69266, 74080, 122062, 117172, 86999, 206214, 86845, 144921, 328452, 213238, 309939, 387079, 652124, 504874, 694074, 596997, 763820, 741672, 404949

Offset: 2

Jon E. Schoenfield, Mar 23 2021

nonn

For n > 42, a(n) > 763820 = a(42).

a(19)=519 corresponds to the noteworthy record high value (19) in A342068 that occurs at n=519.

			5 is the smallest k such that A342068(k)=3, so a(3)=5.

Cf. A342068, A342070, A342071, A342839.

A342852 a(n) = A342068(10^n).

Original entry on oeis.org

2, 5, 5, 7, 8, 12, 33, 52, 93, 236, 479, 1265, 2782, 6650, 15539

Offset: 0

Jon E. Schoenfield, Mar 24 2021

In the table in the Example section, the numbers in the columns for n = 0..9 are the first a(n) terms of A010051, A038800, A038822, ..., A038829, respectively; each column ends at the first term that is greater than the term above it.

a(n) >= a(n-1)/10. - Chai Wah Wu, Mar 25 2021

			Number of primes in the interval [(k-1)*10^n + 1, k*10^n]:
.
   k\n| 0 1  2   3    4    5     6      7       8        9
  ----+---------------------------------------------------
    1 | 0 4 25 168 1229 9592 78498 664579 5761455 50847534
    2 | 1 4 21 135 1033 8392 70435 606028 5317482 47374753
    3 |   2 16 127  983 8013 67883 587252 5173388 46227250
    4 |   2 16 120  958 7863 66330 575795 5084001 45512275
    5 |   3 17 119  930 7678 65367 567480 5019541 44992411
    6 |        114  924 7560 64336 560981 4968836 44591145
    7 |        117  878 7445 63799 555949 4928228 44258984
    8 |             902 7408 63129 551318 4893248 43979302
    9 |                 7323 62712 547572 4863036 43739541
   10 |                 7224 62090 544501 4838319 43529316
   11 |                 7216 61938 541854 4814936 43336106
   12 |                 7224 61543 538339 4792235 43167234
   13 |                      61192 536539 4773628 43014349
   14 |                      60825 534012 4757140 42870136
   15 |                      60627 532197 4741055 42740180
  ... |                        ...    ...     ...      ...
   32 |                      57836 510685 4572164 41368791
   33 |                      57852 510269 4565024 41316074
  ... |                               ...     ...      ...
   51 |                            498385 4475770 40575830
   52 |                            498435 4472349 40547028
  ... |                                       ...      ...
   92 |                                   4357534 39621606
   93 |                                   4360247 39603089
  ... |                                                ...
  235 |                                           38196269
  236 |                                           38197686

Cf. A010051, A038800, A038822, A038823, A038824, A038825, A038826, A038827, A038828, A038829, A342068.

a(10)-a(12) from Chai Wah Wu, Mar 25 2021

a(13)-a(14) from Chai Wah Wu, Mar 31 2021

A342070 Numbers k such that there are more primes in the interval [2k+1, 3k] than there are in the interval [k+1, 2*k].

Original entry on oeis.org

5, 8, 14, 18, 20, 29, 47, 48, 67, 68, 81, 95, 109, 110, 111, 113, 168, 173, 277, 278, 280, 281, 283, 284, 288, 293, 295, 296, 710, 711, 713, 1323

Offset: 1

Jon E. Schoenfield, Mar 23 2021

Conjecture: 1323 is the final term.

If there are at least as many primes in [1, m] as there are in [m+1, 2*m] for all positive integers m, then this sequence consists of the numbers k such that A342068(k)=3.

			The intervals [1, 100], [101, 200], and [201, 300] contain 25, 21, and 16 primes respectively (cf. A038822); 16 < 21, so 100 is not a term of the sequence.
The intervals [1, 20], [21, 40], and [41, 60] contain 8, 4, and 5 primes, respectively; 5 > 4, so 20 is a term.

Cf. A038822, A342068, A342069, A342071, A342839.

from sympy import primepi
def ok(n): return primepi(3*n) > 2*primepi(2*n) - primepi(n)
print([m for m in range(9999) if ok(m)]) # Michael S. Branicky, Mar 23 2021

A342071 Numbers k such that there are more primes in the interval [3k+1, 4k] than there are in the interval [2k+1, 3k].

Original entry on oeis.org

12, 19, 22, 32, 42, 45, 49, 50, 52, 54, 57, 59, 70, 71, 72, 73, 74, 75, 101, 102, 115, 116, 117, 121, 122, 123, 124, 126, 132, 143, 180, 182, 184, 185, 186, 187, 188, 189, 190, 192, 194, 195, 197, 268, 269, 309, 310, 311, 312, 322, 323, 325, 326, 327, 328, 329

Offset: 1

Jon E. Schoenfield, Mar 23 2021

nonn

After a(194)=3977, there are no more terms < 100000.
Conjecture: a(194)=3977 is the final term.
For each of the first 194 terms k, there are at least as many primes in [1, k] as there are in [k+1, 2*k], and at least as many primes in [k+1, 2*k] as there are in [2*k+1, 3*k], so A342068(k)=4.

			The intervals [1, 100], [101, 200], [201, 300], and [301, 400] contain 25, 21, 16, and 16 primes respectively (cf. A038822); the 4th interval does not contain more primes than does the 3rd, so 100 is not a term of the sequence.
However, the intervals [1, 101], [102, 202], [203, 303], and [304, 404] contain 26, 20, 16, and 17 primes, respectively; 17 > 16, so 101 is a term.

Cf. A342068, A342069, A342070, A342839.

A342839 Numbers k such that there are more primes in the interval [4k+1, 5k] than there are in the interval [3k+1, 4k].

Original entry on oeis.org

1, 4, 7, 9, 10, 15, 16, 22, 23, 24, 25, 34, 36, 37, 39, 40, 47, 55, 56, 57, 58, 64, 67, 82, 84, 86, 87, 88, 91, 93, 94, 95, 96, 97, 98, 99, 100, 102, 104, 105, 106, 107, 130, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 144, 146, 147, 148, 149, 150, 153

Offset: 1

Jon E. Schoenfield, Mar 23 2021

nonn

After a(876) = 11895, there are no more terms < 100000.
Conjecture: a(876) = 11895 is the final term.
There exist eight terms k for which A342068(k) != 5: A342068(k) = 2 for k = 1; A342068(k) = 3 for k = 47, 67, 95, and 1323; and A342068(k)=4 for k = 22, 57, and 102.

			The intervals [1, 100], [101, 200], [201, 300], [301, 400], and [401, 500] contain 25, 21, 16, 16, and 17 primes, respectively (cf. A038822); 17 > 16, so 100 is a term of the sequence.

Cf. A342068, A342069, A342070, A342071.

cp's OEIS Frontend

A342069 a(n) is the smallest k such that A342068(k) = n.

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A342852 a(n) = A342068(10^n).

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A342070 Numbers k such that there are more primes in the interval [2k+1, 3k] than there are in the interval [k+1, 2*k].

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A342071 Numbers k such that there are more primes in the interval [3k+1, 4k] than there are in the interval [2k+1, 3k].

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A342839 Numbers k such that there are more primes in the interval [4k+1, 5k] than there are in the interval [3k+1, 4k].

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cp's OEIS Frontend

A342069 a(n) is the smallest k such that A342068(k) = n.

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A342852 a(n) = A342068(10^n).

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A342070 Numbers k such that there are more primes in the interval [2*k+1, 3*k] than there are in the interval [k+1, 2*k].

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A342071 Numbers k such that there are more primes in the interval [3*k+1, 4*k] than there are in the interval [2*k+1, 3*k].

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A342839 Numbers k such that there are more primes in the interval [4*k+1, 5*k] than there are in the interval [3*k+1, 4*k].

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A342070 Numbers k such that there are more primes in the interval [2k+1, 3k] than there are in the interval [k+1, 2*k].

A342071 Numbers k such that there are more primes in the interval [3k+1, 4k] than there are in the interval [2k+1, 3k].

A342839 Numbers k such that there are more primes in the interval [4k+1, 5k] than there are in the interval [3k+1, 4k].