A356221 -id:A356221

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A356223

Position of n-th appearance of 2n in the sequence of prime gaps (A001223). If 2n does not appear at least n times, set a(n) = -1.

+10
4

2, 6, 15, 79, 68, 121, 162, 445, 416, 971, 836, 987, 2888, 1891, 1650, 5637, 5518, 4834, 9237, 8152, 10045, 21550, 20248, 20179, 29914, 36070, 24237, 53355, 52873, 34206, 103134, 90190, 63755, 147861, 98103, 117467, 209102, 206423, 124954, 237847, 369223

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Prime gaps (A001223) are the differences between consecutive prime numbers. They begin: 1, 2, 2, 4, 2, 4, 2, 4, 6, ...

LINKS

Table of n, a(n) for n=1..41.

EXAMPLE

We need the first 15 prime gaps (1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6) before we reach the 3rd appearance of 6, so a(6) = 15.

MATHEMATICA

nn=1000;

gaps=Differences[Array[Prime, nn]];

Table[Position[gaps, 2*n][[n, 1]], {n, Select[Range[nn], Length[Position[gaps, 2*#]]>=#&]}]

CROSSREFS

The first appearances are at A038664, seconds A356221.

Diagonal of A356222.

A001223 lists the prime gaps.

A073491 lists numbers with gapless prime indices.

A356224 counts divisors with gapless prime indices, complement A356225.

A356226 = gapless interval lengths of prime indices, run-lengths A287170.

Cf. A000005, A028334, A029709, A060681, A119313, A137921, A193829, A274121.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 04 2022

STATUS

approved

A356222

Array read by antidiagonals upwards where A(n,k) is the position of the k-th appearance of 2n in the sequence of prime gaps A001223. If A001223 does not contain 2n at least k times, set A(n,k) = -1.

+10
2

2, 4, 3, 9, 6, 5, 24, 11, 8, 7, 34, 72, 15, 12, 10, 46, 42, 77, 16, 14, 13, 30, 47, 53, 79, 18, 19, 17, 282, 62, 91, 61, 87, 21, 22, 20, 99, 295, 66, 97, 68, 92, 23, 25, 26, 154, 180, 319, 137, 114, 80, 94, 32, 27, 28, 189, 259, 205, 331, 146, 121, 82, 124, 36, 29, 33

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Prime gaps (A001223) are the differences between consecutive prime numbers. They begin: 1, 2, 2, 4, 2, 4, 2, 4, 6, ...

This is a permutation of the positive integers > 1.

LINKS

Table of n, a(n) for n=1..66.

EXAMPLE

Array begins:

k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9

n=1: 2 3 5 7 10 13 17 20 26

n=2: 4 6 8 12 14 19 22 25 27

n=3: 9 11 15 16 18 21 23 32 36

n=4: 24 72 77 79 87 92 94 124 126

n=5: 34 42 53 61 68 80 82 101 106

n=6: 46 47 91 97 114 121 139 168 197

n=7: 30 62 66 137 146 150 162 223 250

n=8: 282 295 319 331 335 378 409 445 476

n=9: 99 180 205 221 274 293 326 368 416

For example, the positions in A001223 of appearances of 2*3 begin: 9, 11, 15, 16, 18, 21, 23, ..., which is row n = 3 (A320701).

MATHEMATICA

gapa=Differences[Array[Prime, 10000]];

Table[Position[gapa, 2*(k-n+1)][[n, 1]], {k, 6}, {n, k}]

CROSSREFS

The row containing n is A028334(n).

Row n = 1 is A029707.

Row n = 2 is A029709.

Column k = 1 is A038664.

The column containing n is A274121(n).

Column k = 2 is A356221.

The diagonal A(n,n) is A356223.

A001223 lists the prime gaps.

A073491 lists numbers with gapless prime indices.

A356224 counts even divisors with gapless prime indices, complement A356225.

Cf. A066205, A119313, A193829, A287170, A328457, A356226, A356232.