A270189 -id:A270189 - cp's OEIS frontend

A270201 Permutation of natural numbers: a(1) = 1, a(A270189(1+n)) = 2 * a(n), a(A270190(n)) = 1 + 2*a(n).

1, 2, 4, 8, 16, 32, 64, 128, 3, 256, 5, 6, 512, 10, 9, 17, 12, 33, 1024, 20, 65, 18, 129, 34, 24, 66, 2048, 40, 130, 36, 258, 257, 68, 48, 132, 7, 513, 4096, 11, 13, 80, 260, 72, 516, 514, 1025, 21, 136, 96, 264, 19, 14, 1026, 35, 25, 67, 8192, 2049, 22, 26, 160, 520, 144, 1032, 1028, 2050, 41, 42, 272, 192, 131, 528, 37, 259, 38, 69, 28

Offset: 1

Antti Karttunen, Mar 16 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A270202.
Cf. A137264, A269849, A269850, A270189, A270190.
Similar permutation: A270193.

a(1) = 1, for n > 1, if A137264(n) = 0 [when n is in A270190], a(n) = 1 + 2*a(A269850(n)), otherwise a(n) = 2 * a(A269849(n)-1).

A270202 Permutation of natural numbers: a(1) = 1, a(2n) = A270189(1+a(n)), a(2n+1) = A270190(a(n)).

Original entry on oeis.org

1, 2, 9, 3, 11, 12, 36, 4, 15, 14, 39, 17, 40, 52, 108, 5, 16, 22, 51, 20, 47, 59, 114, 25, 55, 60, 118, 77, 167, 156, 312, 6, 18, 24, 54, 30, 73, 75, 165, 28, 67, 68, 139, 85, 185, 166, 339, 34, 84, 80, 174, 87, 187, 173, 347, 117, 227, 254, 496, 236, 461, 475, 852, 7, 21, 26, 56, 33, 76, 79, 170, 43, 99, 112, 216, 115, 219, 251, 490, 41

Offset: 1

Antti Karttunen, Mar 16 2016

This sequence can be represented as a binary tree. Each left hand child is produced as A270189(1+n), and each right hand child as A270190(n), when the parent node contains n:
                                    1
                 ................../ \..................
                2                                       9
      3......../ \........11                 12......../ \........36
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  4       15         14       39         17       40         52      108
5  16   22  51     20  47   59  114    25  55   60  118    77  167  156 312
etc.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A270201.
Cf. A270189, A270190.
Similar permutation: A270194.

a(1) = 1, a(2n) = A270189(1+a(n)), a(2n+1) = A270190(a(n)).

A270190 Numbers n for which prime(n+1)-prime(n) is a multiple of three.

Original entry on oeis.org

9, 11, 15, 16, 18, 21, 23, 32, 36, 37, 39, 40, 46, 47, 51, 54, 55, 56, 58, 67, 71, 73, 74, 76, 84, 86, 91, 96, 97, 99, 100, 102, 103, 105, 107, 108, 110, 111, 114, 118, 119, 121, 123, 129, 130, 133, 139, 160, 161, 164, 165, 167, 168, 170, 174, 179, 180, 184, 185, 187, 188, 194, 195, 197, 199, 200, 202, 203, 205, 208, 210

Offset: 1

Antti Karttunen, Mar 16 2016

nonn

Numbers n for which A001223(n) = 0 modulo 3.
See comments in A270189 and A269364.
Equivalently, numbers n for which prime(n+1)-prime(n) is a multiple of six. See A276414 for runs of increasing length of consecutive integers. - M. F. Hasler, Sep 03 2016

			9 is present as the difference between A000040(9+1) = 29 and A000040(9) = 23 is 6, a multiple of three.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Terence Tao, Biases between consecutive primes, blog entry March 14, 2016

Complement: A270189.
Positions of zeros in A137264.
Left inverse: A269850.
Cf. also A001223, A269364, A270191, A270192, A269399.

Select[Range@ 210, Divisible[Prime[# + 1] - Prime@ #, 3] &] (* Michael De Vlieger, Mar 17 2016 *)
PrimePi/@Select[Partition[Prime[Range[350]],2,1],Divisible[#[[2]]-#[[1]], 3]&][[All,1]] (* Harvey P. Dale, Jul 11 2017 *)

isok(n) = ((prime(n+1) - prime(n)) % 3) == 0; \\ Michel Marcus, Mar 17 2016

Other identities. For all n >= 1:
a(n) = A269399(n) + 6.
A269850(a(n)) = n.

A137264 Prime number gaps read modulo 3.

Original entry on oeis.org

1, 2, 2, 1, 2, 1, 2, 1, 0, 2, 0, 1, 2, 1, 0, 0, 2, 0, 1, 2, 0, 1, 0, 2, 1, 2, 1, 2, 1, 2, 1, 0, 2, 1, 2, 0, 0, 1, 0, 0, 2, 1, 2, 1, 2, 0, 0, 1, 2, 1, 0, 2, 1, 0, 0, 0, 2, 0, 1, 2, 1, 2, 1, 2, 1, 2, 0, 1, 2, 1, 0, 2, 0, 0, 1, 0, 2, 1, 2, 1, 2, 1, 2, 0, 1, 0, 2, 1, 2, 1, 0, 2, 1, 2, 1, 0, 0, 2, 0, 0, 1, 0

Offset: 1

Noel H. Patson (n.patson(AT)cqu.edu.au), Mar 12 2008

Conjecture: The only digit that is repeated in the sequence is 0 except for n=2 and n=3 where 2 repeats. So 1 may be followed by 2 or 0; 2 may be followed by 1 or 0; 0 may be followed by 0 or 1 or 2. this has been confirmed for the first million prime gaps.
The conjecture is true, because any three numbers whose differences are (1, 1) or (2, 2) will form a complete residue system modulo 3, and hence one of them will be a multiple of 3. - Karl W. Heuer, Mar 16 2016
See comments at A269364. - Antti Karttunen, Mar 17 2016

Terence Tao, Biases between consecutive primes, blog entry March 14, 2016

Cf. A001223.

Cf. also A269364, A270189, A270190, A270191, A270192.

n=1000;(*The length of the list*) Mod[Differences[Table[Prime[i], {i, n}]], 3]

(define (A137264 n) (modulo (A001223 n) 3)) ;; Antti Karttunen, Mar 16 2016

A269364 Difference between the number of occurrences of prime gaps not divisible by 3, versus number of prime gaps that are multiples of 3, up to n-th prime gap: a(n) = A269849(n) - A269850(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 7, 8, 7, 8, 9, 10, 9, 8, 9, 8, 9, 10, 9, 10, 9, 10, 11, 12, 13, 14, 15, 16, 17, 16, 17, 18, 19, 18, 17, 18, 17, 16, 17, 18, 19, 20, 21, 20, 19, 20, 21, 22, 21, 22, 23, 22, 21, 20, 21, 20, 21, 22, 23, 24, 25, 26, 27, 28, 27, 28, 29, 30, 29, 30, 29, 28, 29, 28, 29, 30, 31, 32, 33, 34, 35, 34

Offset: 1

Antti Karttunen, Mar 17 2016

nonn

This is related to "Lemke Oliver-Soundararajan bias", term first used by Terence Tao March 14, 2016 in his blog.

Antti Karttunen, Table of n, a(n) for n = 1..50000
Robert J. Lemke Oliver and Kannan Soundararajan, Unexpected biases in the distribution of consecutive primes, arXiv:1603.03720 [math.NT], 2016.
Terence Tao, Biases between consecutive primes, blog entry March 14, 2016

Cf. A001223, A137264, A269849, A269850, A270189, A270190.

Cf. also A270310, A038698.

a(n) = sum(k=1, n, ((prime(k+1) - prime(k)) % 3) != 0) - sum(k=1, n, ((prime(k+1) - prime(k)) % 3) == 0); \\ Michel Marcus, Mar 18 2016

(define (A269364 n) (- (A269849 n) (A269850 n)))

a(n) = A269849(n) - A269850(n).

A269389 Numbers n for which prime(n+7)-prime(n+6) is not a multiple of three.

Original entry on oeis.org

1, 2, 4, 6, 7, 8, 11, 13, 14, 16, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 32, 35, 36, 37, 38, 39, 42, 43, 44, 46, 47, 51, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 66, 69, 71, 72, 73, 74, 75, 76, 77, 79, 81, 82, 83, 84, 86, 87, 88, 89, 92, 95, 98, 100, 103, 106, 107, 109, 110, 111, 114, 116, 118, 119, 120, 121, 122, 125

Offset: 1

Antti Karttunen, Mar 16 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Complement: A269399.
Left inverse: A269362.
Cf. A270189.
Cf. also A270199.

Select[Range@ 125, ! Divisible[Prime[# + 7] - Prime[# + 6], 3] &] (* Michael De Vlieger, Mar 18 2016 *)
Drop[Flatten[Position[Partition[Prime[Range[200]],2,1],?(Mod[#[[2]]- #[[1]],3]!=0&),1,Heads->False]],6]-6 (* _Harvey P. Dale, Dec 17 2018 *)

isok(n) = ((prime(n+7) - prime(n+6)) % 3) != 0; \\ Michel Marcus, Mar 18 2016

a(n) = A270189(6+n) - 6.
Other identities. For all n >= 1:
A269362(a(n)) = n.

A269849 a(n) = number of integers k <= n for which prime(k+1)-prime(k) is not a multiple of three.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 8, 9, 9, 10, 11, 12, 12, 12, 13, 13, 14, 15, 15, 16, 16, 17, 18, 19, 20, 21, 22, 23, 24, 24, 25, 26, 27, 27, 27, 28, 28, 28, 29, 30, 31, 32, 33, 33, 33, 34, 35, 36, 36, 37, 38, 38, 38, 38, 39, 39, 40, 41, 42, 43, 44, 45, 46, 47, 47, 48, 49, 50, 50, 51, 51, 51, 52, 52, 53, 54, 55, 56, 57, 58, 59, 59, 60

Offset: 1

Antti Karttunen, Mar 16 2016

nonn

a(n) = number of terms of A270189 <= n, the least monotonic left inverse of A270189.

See comments at A269364.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Terence Tao, Biases between consecutive primes, blog entry March 14, 2016

Cf. A270189, A269850, A269362, A269364.

Table[Count[Select[Range@ 125, Mod[Prime[# + 1] - Prime@ #, 3] != 0 &], k_ /; k <= n], {n, 85}] (* Michael De Vlieger, Mar 17 2016 *)
Accumulate[If[Mod[#,3]==0,0,1]&/@Differences[Prime[Range[90]]]] (* Harvey P. Dale, Apr 15 2024 *)

a(n) = sum(k=1, n, ((prime(k+1) - prime(k)) % 3) != 0); \\ Michel Marcus, Mar 18 2016

Other identities. For all n >= 1:

a(A270189(n)) = n.

A270191 Numbers n for which (prime(n+1)-prime(n)) mod 3 = 1.

Original entry on oeis.org

1, 4, 6, 8, 12, 14, 19, 22, 25, 27, 29, 31, 34, 38, 42, 44, 48, 50, 53, 59, 61, 63, 65, 68, 70, 75, 78, 80, 82, 85, 88, 90, 93, 95, 101, 106, 112, 115, 117, 122, 125, 127, 131, 134, 136, 138, 141, 143, 145, 147, 149, 151, 153, 155, 157, 159, 163, 169, 172, 175, 177, 181, 183, 189, 191, 193, 198, 204, 207, 211, 213, 217, 222

Offset: 1

Antti Karttunen, Mar 16 2016

nonn

			1 is present as prime(2)-prime(1) = 3-2 = 1 = 1 mod 3.
4 is present as prime(5)-prime(4) = 11-7 = 4 = 1 mod 3.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Subsequence of A270189.
Positions of ones in A137264.
Cf. A000040, A001223, A270190, A270192.

Select[Range@ 222, Mod[Prime[# + 1] - Prime@ #, 3] == 1 &] (* Michael De Vlieger, Mar 17 2016 *)

isok(n) = ((prime(n+1) - prime(n)) % 3) == 1; \\ Michel Marcus, Mar 17 2016

A270192 Numbers n for which (prime(n+1)-prime(n)) mod 3 = 2.

Original entry on oeis.org

2, 3, 5, 7, 10, 13, 17, 20, 24, 26, 28, 30, 33, 35, 41, 43, 45, 49, 52, 57, 60, 62, 64, 66, 69, 72, 77, 79, 81, 83, 87, 89, 92, 94, 98, 104, 109, 113, 116, 120, 124, 126, 128, 132, 135, 137, 140, 142, 144, 146, 148, 150, 152, 154, 156, 158, 162, 166, 171, 173, 176, 178, 182, 186, 190, 192, 196, 201, 206, 209, 212, 215, 220, 223, 225

Offset: 1

Antti Karttunen, Mar 16 2016

nonn

			2 is present as prime(3) - prime(2) = 5 - 3 = 2 = 2 modulo 3.
24 is present as prime(24) = 89, prime(25) = 97 and 97-89 = 8 = 2 modulo 3.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Subsequence of A270189.
Positions of 2's in A137264.
Cf. A000040, A001223, A270190, A270191.
Differs from its subsequence A029707 for the first time at n=9.

Select[Range@ 225, Mod[Prime[# + 1] - Prime@ #, 3] == 2 &] (* Michael De Vlieger, Mar 17 2016 *)
Position[Differences[Prime[Range[300]]],?(Mod[#,3]==2&)]//Flatten (* _Harvey P. Dale, Jul 22 2016 *)

isok(n) = ((prime(n+1) - prime(n)) % 3) == 2; \\ Michel Marcus, Mar 17 2016

cp's OEIS Frontend

A270201 Permutation of natural numbers: a(1) = 1, a(A270189(1+n)) = 2 * a(n), a(A270190(n)) = 1 + 2*a(n).

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A270202 Permutation of natural numbers: a(1) = 1, a(2n) = A270189(1+a(n)), a(2n+1) = A270190(a(n)).

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A270190 Numbers n for which prime(n+1)-prime(n) is a multiple of three.

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A137264 Prime number gaps read modulo 3.

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A269364 Difference between the number of occurrences of prime gaps not divisible by 3, versus number of prime gaps that are multiples of 3, up to n-th prime gap: a(n) = A269849(n) - A269850(n).

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A269389 Numbers n for which prime(n+7)-prime(n+6) is not a multiple of three.

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A269849 a(n) = number of integers k <= n for which prime(k+1)-prime(k) is not a multiple of three.

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A270191 Numbers n for which (prime(n+1)-prime(n)) mod 3 = 1.

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