A047948 -id:A047948 - cp's OEIS frontend

A219096 Indices of primes p such that the next two larger primes are p+6 and p+12.

15, 36, 39, 54, 55, 73, 102, 107, 110, 118, 129, 160, 164, 184, 187, 194, 199, 218, 271, 272, 291, 339, 358, 387, 419, 426, 464, 465, 508, 520, 553, 605, 621, 629, 667, 682, 683, 702, 709, 710, 733, 761, 791, 813, 821, 822, 829, 830, 882, 896, 952, 962, 988

Offset: 1

Clark Kimberling, Mar 05 2013

The primes themselves are given by A047948. Conjecture: if k == 0 mod 6 then there exists a prime p such that p-k, p, p+k are consecutive primes. (This would follow from a proof of Dickson's conjecture; see the Comments and References at A186311.)

			a(1) = 15 since p(15), p(16), p(17) are consecutive primes (47, 53, 59) with common difference 6: 53 - 47 = 6, and 59 - 53 = 6.

Clark Kimberling, Table of n, a(n) for n = 1..10000
Consecutive primes in arithmetic progression

Cf. A047948, A128940, A000720.

z = 1000; t = Differences[Prime[Range[z]]];
f[n_] := If[t[[n + 1]] - t[[n]] == 0, t[[n]], 0]
u = Table[f[n], {n, 1, z - 2}]; s = Flatten[Position[u, 6]] (*A219096*)
Prime[s]  (*A047948*)
Flatten[Position[Partition[Prime[Range[1000]],3,1],?(Differences[#] == {6,6}&),{1},Heads->False]] (* _Harvey P. Dale, Jan 01 2015 *)

n=0;p=2;q=3;forprime(r=5,1e6,n++;if(r-p==12&&q-p==6,print1(n", "));p=q;q=r) \\ Charles R Greathouse IV, Mar 05 2013

a(n) = A000720(A047948(n)). - M. F. Hasler, Mar 11 2013

A224325 First of three consecutive primes in arithmetic progression with gap of 6n, and such that a(n) > a(n-1).

Original entry on oeis.org

47, 199, 20183, 40039, 69593, 255767, 689467, 3565931, 6314393, 9113263, 12012677, 23346737, 43607351, 69266033, 75138781, 324237847, 460475467, 652576321, 742585183, 747570079, 807620651, 2988119207, 12447231761

Offset: 1

M. F. Hasler, Apr 03 2013

nonn

Without the condition on monotonicity, this would be essentially the same as A052187, but there 255767 is followed by 247099, while monotonicity here gives 689467. Similarly, following a(9) = A052187(10) = 6314393 we have a(10) = 9113263, while A052187(11) = 4911251. The next term which is not matching is a(14) = 69266033 vs A052187(15) = 34346203. One may notice that the two terms differ approximately by a factor of 2.

			a(1) = A047948(1) = 47 is the least prime p(k) such that p(k+1) - p(k) = p(k+2) - p(k+1) = 6.
a(2) = A052188(1) = 199 is the least prime p(k) > 47 such that p(k+1) - p(k) = p(k+2) - p(k+1) = 12.
a(3) = A052189(1) = 20183 is the least prime p(k) > 199 such that p(k+1) - p(k) = p(k+2) - p(k+1) = 18.
a(4) = A052190(1) = 40039 is the least prime p(k) > 20183 such that p(k+1) - p(k) = p(k+2) - p(k+1) = 24.
a(5) = A052195(1) = 69593 is the least prime p(k) > 40039 such that p(k+1) - p(k) = p(k+2) - p(k+1) = 30.

Pierre CAMI, Table of n, a(n) for n = 1..23

Cf. A224324 (gaps of 30n).

g=6;o=2;forprime(p=2,,o+g==(o=p)||next;nextprime(p+1)==p+g||next;print1(p-g",");g+=6)

A329578 First of three consecutive primes with common gap 48.

Original entry on oeis.org

3565931, 3653863, 3985903, 5425613, 5647361, 6126971, 6292081, 6532553, 7133983, 7360363, 7389493, 7700131, 7865833, 7956163, 8467903, 8708291, 8972701, 9203743, 9603361, 9863551, 10279813, 10971743, 11998391, 12225251, 12474251, 12620843, 12966881, 13288211, 13376261, 13543451

Offset: 1

M. F. Hasler, Jan 02 2020

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
OEIS Wiki, Consecutive primes in arithmetic progression, updated Jan. 2020.

Subsequence of A134123 (first of two primes with common gap 48).
A067388 (first of four primes with common gap 48) is a subsequence.
Cf. A047948, A052188, A052189, A052190, A052195, A052197, A052198, A089234 (analog for gaps 2, 4, 6, 12, 18, 24, ..., 60).

[p:p in PrimesUpTo(14000000)| NextPrime(p)-p eq 48 and NextPrime(p+48)-p eq 96]; // Marius A. Burtea, Jan 03 2020

Select[Partition[Prime[Range[900000]],3,1],Differences[#]=={48,48}&] [[All,1]] (* Harvey P. Dale, Aug 23 2021 *)

vecextract( A134123, select(t->t==48, A134123[^1]-A134123[^-1], 1)) \\ Terms of A134123 with indices corresponding to first differences of 48: gives a(1..56) from A134123(1..10^4).

A280201 Let the smallest of three successive primes p, p+d, p+2d be a so-called d-triple and b(n) the sequence of d-triples with d<>6. Then a(n) is the number of 6-triples between b(n) and b(n+1).

Original entry on oeis.org

3, 15, 13, 3, 19, 5, 4, 0, 1, 8, 8, 13, 0, 4, 2, 2, 1, 5, 0, 2, 0, 1, 0, 1, 0, 1, 1, 4, 5, 1, 1, 8, 3, 1, 1, 3, 3, 2, 4, 2, 2, 2, 0, 1, 2, 5, 1, 1, 2, 2

Offset: 1

Gerhard Kirchner, Dec 28 2016

nonn

The sequence of all d-triples A122535(n) = (3), 47, 151, 167, (199), 251, 257, 367, 557, 587, 601, 647, 727, 941, 971, 1097, 1117, 1181, 1217, 1361, (1499), ... is the union of A047948(n) with 6-triples and b(n) with terms in brackets. There are three 6-triples between 3 and 199 and 15 6-triples between 199 and 1499. Thus a(1)=3 (see example) and a(2)=15.
The average of the first 10 terms is (3+15+13+3+19+5+4+0+1+8)/10 = 7.1. This means that, in this section, the 6-triples are more than 7 times as frequent as the other d-triples as a whole. Let us compare longer sections of a(n) with different magnitudes of n, for example (with S(n)=sum(a(k),k,1,n)/n): n <= 10000 100000 733158
                          S(n) =  1.28   0.98   0.81
n=733158 was the largest available index when I analyzed a pool of primes <=10^9.
Result: For small n, 6-triples are more frequent than the whole of other d-triples; for large n, the reverse is true. Does S(n) tend to zero? It seems so, see link "Tendency of a(n)". - Gerhard Kirchner, Dec 28 2016

			The first d-triples are 3 (,5,7, d=2); 47 (,53,59, d=6); 151 (,157,163, d=6); 167 (,173,179, d=6); 199 (,211,223, d=12). So there are three 6-triples between the 2-triple and the 12-triple: a(1)=3.

Gerhard Kirchner, Table of n, a(n) for n = 1..10000
Gerhard Kirchner, Tendency of a(n)

Cf. A047948, A122535.

A294147 Initial member of 9 consecutive primes {a, b, c, d, e, f, g, h, i} such that (a + b + c)/3, (d + e + f)/3 and (g + h + i)/3 are all prime.

Original entry on oeis.org

63487, 462067, 830777, 847507, 1012159, 1049773, 1250611, 1268747, 1372537, 1372559, 1589657, 1988237, 2567557, 2696569, 2874673, 2967317, 3676111, 3718657, 4196987, 4255067, 4550867, 4669333, 5217911, 5225147, 5716031, 6019553, 6103171, 6725657, 6725731, 7143557

Offset: 1

K. D. Bajpai, Oct 23 2017

nonn

			63487 is a term because it is the initial term of 9 consecutive primes {63487, 63493, 63499, 63521, 63527, 63533, 63541, 63559, 63577} = {a, b, c, d, e, f, g, h, i}: the arithmetic mean of three sets, i.e., (a + b + c)/ 3, (d + e + f)/3 and (g + h + i)/3 is prime.

Cf. A000040, A047948, A122535, A293393, A293395.

Select[Partition[Prime@ Range[5*10^5], 9, 1], Function[{a, b, c, d, e, f, g, h, i}, AllTrue[{(a + b + c)/3, (d + e + f)/3, (g + h + i)/3}, PrimeQ]] @@ # &][[All, 1]] (* Michael De Vlieger, Oct 23 2017 *)

A374719 Primes p such that p + 48 and p + 96 are also prime.

Original entry on oeis.org

5, 11, 13, 31, 41, 53, 61, 83, 101, 103, 131, 181, 263, 283, 353, 383, 461, 521, 523, 613, 643, 661, 691, 761, 811, 881, 991, 1013, 1021, 1153, 1181, 1201, 1231, 1483, 1511, 1523, 1531, 1571, 1693, 1783, 1901, 1931, 2083, 2293, 2341, 2351, 2671, 2693, 2741

Offset: 1

James S. DeArmon, Jul 17 2024

nonn

			5 is a term because 5, 5+48, and 5+96 are all prime.

Cf. A001223, A033451, A047948, A052188, A052189, A052190, A279765.

is(n)=isprime(n+96) && isprime(n+48) && isprime(n) \\ Charles R Greathouse IV, Jul 25 2024

a(n) >> n log^3 n. - Charles R Greathouse IV, Jul 25 2024

cp's OEIS Frontend

A219096 Indices of primes p such that the next two larger primes are p+6 and p+12.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A224325 First of three consecutive primes in arithmetic progression with gap of 6n, and such that a(n) > a(n-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A329578 First of three consecutive primes with common gap 48.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A280201 Let the smallest of three successive primes p, p+d, p+2d be a so-called d-triple and b(n) the sequence of d-triples with d<>6. Then a(n) is the number of 6-triples between b(n) and b(n+1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A294147 Initial member of 9 consecutive primes {a, b, c, d, e, f, g, h, i} such that (a + b + c)/3, (d + e + f)/3 and (g + h + i)/3 are all prime.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A374719 Primes p such that p + 48 and p + 96 are also prime.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula