A269849 -id:A269849 - cp's OEIS frontend

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).

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).

A270189 Numbers n for which (prime(n+1)-prime(n)) is not a multiple of three.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 13, 14, 17, 19, 20, 22, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 38, 41, 42, 43, 44, 45, 48, 49, 50, 52, 53, 57, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 72, 75, 77, 78, 79, 80, 81, 82, 83, 85, 87, 88, 89, 90, 92, 93, 94, 95, 98, 101, 104, 106, 109, 112, 113, 115, 116, 117, 120, 122, 124, 125

Offset: 1

Antti Karttunen, Mar 16 2016

nonn

See A269364 for the effect of the bias that favors these terms over the terms of A270190.

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

Complement: A270190.
Disjoint union of A270191 and A270192.
Positions of 1's and 2's in A137264.
Left inverse: A269849.
Cf. A000040, A001223, A269364, A269389.

Select[Range@ 125, Mod[Prime[# + 1] - Prime@ #, 3] != 0 &] (* Michael De Vlieger, Mar 17 2016 *)

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

Other identities. For all n >= 1:

A269849(a(n)) = n.

A269362 Least monotonic left inverse of A269389.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Mar 16 2016

nonn

a(n) = number of terms of A269389 <= n.

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

Cf. A269389, A269849.

Cf. also A270193, A270199.

a(n) = A269849(6+n) - 6.
Other identities. For all n >= 1:
a(A269389(n)) = n.

A269850 a(n) = number of integers k <= n for which prime(k+1)-prime(k) is a multiple of three.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 3, 4, 4, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 10, 10, 11, 12, 12, 12, 12, 12, 12, 13, 14, 14, 14, 14, 15, 15, 15, 16, 17, 18, 18, 19, 19, 19, 19, 19, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 22, 23, 23, 24, 24, 24, 24, 24, 24, 24, 24, 25, 25, 26, 26, 26, 26, 26, 27

Offset: 1

Antti Karttunen, Mar 16 2016

nonn

a(n) = number of terms of A270190 <= n, the least monotonic left inverse of A270190.

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. A270190, A269849, A269364.

Table[Count[Select[Range@ 125, Divisible[Prime[# + 1] - Prime@ #, 3] &], k_ /; k <= n], {n, 91}] (* Michael De Vlieger, Mar 17 2016 *)

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(A270190(n)) = n.

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

Original entry on oeis.org

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).

cp's OEIS Frontend

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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A270189 Numbers n for which (prime(n+1)-prime(n)) is not a multiple of three.

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A269362 Least monotonic left inverse of A269389.

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A269850 a(n) = number of integers k <= n for which prime(k+1)-prime(k) is a multiple of three.

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Formula

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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