A270192 -id:A270192 - cp's OEIS frontend

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

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

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.

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

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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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A270189 Numbers n for which (prime(n+1)-prime(n)) 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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