A052190 -id:A052190 - cp's OEIS frontend

A122535 Smallest prime of a triple of successive primes, where the middle one is the arithmetic mean of the other two.

3, 47, 151, 167, 199, 251, 257, 367, 557, 587, 601, 647, 727, 941, 971, 1097, 1117, 1181, 1217, 1361, 1499, 1741, 1747, 1901, 2281, 2411, 2671, 2897, 2957, 3301, 3307, 3631, 3727, 4007, 4397, 4451, 4591, 4651, 4679, 4987, 5101, 5107, 5297, 5381, 5387

Offset: 1

Miklos Kristof, Sep 18 2006

nonn

Subsets are A047948, A052188, A052189, A052190, A052195, A052197, A052198, etc. - R. J. Mathar, Apr 11 2008
Could be generated by searching for cases A001223(i) = A001223(i+1), writing down A000040(i). - R. J. Mathar, Dec 20 2008
a(n) = A006562(n) - A117217(n). - Zak Seidov, Feb 12 2013
These are primes for which the subsequent prime gaps are equal, so (p(k+2)-p(k+1))/(p(k+1)-p(k)) = 1. It is conjectured that prime gaps ratios equal to one are less frequent than those equal to 1/2, 2, 3/2, 2/3, 1/3 and 3. - Andres Cicuttin, Nov 07 2016

			The prime 7 is not in the list, because in the triple (7, 11, 13) of successive primes, 11 is not equal (7 + 13)/2 = 10.
The second term, 47, is the first prime in the triple (47, 53, 59) of primes, where 53 is the mean of 47 and 59.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primes in arithmetic progressions

Cf. A006562, A062839, A102552, A117217, A181424.

a122535 = a000040 . a064113  -- Reinhard Zumkeller, Jan 20 2012

Clear[d2, d1, k]; d2[n_] = Prime[n + 2] - 2*Prime[n + 1] + Prime[n]; d1[n_] = Prime[n + 1] - Prime[n]; k[n_] = -d2[n]/(1 + d1[n])^(3/2); Flatten[Table[If[k[n] == 0, Prime[n], {}], {n, 1, 1000}]] (* Roger L. Bagula, Nov 13 2008 *)
Transpose[Select[Partition[Prime[Range[750]], 3, 1], #[[2]] == (#[[1]] + #[[3]])/2 &]][[1]]  (* Harvey P. Dale, Jan 09 2011 *)

A122535()={n=3;ctr=0;while(ctr<50, avgg=( prime(n-2)+prime(n) )/2;
if( prime(n-1) ==avgg, ctr+=1;print( ctr,"  ",prime(n-2) )  );n+=1); } \\ Bill McEachen, Jan 19 2015

{A000040(i): A000040(i+1)= (A000040(i)+A000040(i+2))/2 }. - R. J. Mathar, Dec 20 2008

a(n) = A000040(A064113(n)). - Reinhard Zumkeller, Jan 20 2012

More terms from Roger L. Bagula, Nov 13 2008

Definition rephrased by R. J. Mathar, Dec 20 2008

A052195 Primes p such that p, p+30, p+60 are consecutive primes.

Original entry on oeis.org

69593, 110651, 134609, 228647, 237791, 250889, 303157, 318919, 396449, 421913, 498271, 507431, 535243, 554317, 629623, 642427, 642457, 668243, 692161, 716003, 729791, 780523, 782581, 790897, 801217, 825131, 829289, 847393, 892291, 902873, 940097, 942449, 963913, 995243, 1027067

Offset: 1

Labos Elemer, Jan 28 2000

nonn

			69593, 69623, 69653 are consecutive primes with equal distance d = 30.
110651, 110681 and 110711 are consecutive primes with equal distance d = 30.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Zak Seidov)
OEIS wiki, Consecutive primes in arithmetic progression: CPAP with given gap, updated Jan. 2020.

Subsequence of A124596 (primes followed by gap 30).
Cf. A047948 (analog for gap 6), A052188 (gap 12), A052189 (gap 18), A052190 (gap 24), A053075 (a(n) + 30).
Cf. A001223 (gaps), A052243 (quadruplets with gap 30), A033451 (quadruplets with gap 6).

Select[Partition[Prime[Range[80000]],3,1],Differences[#]=={30,30}&][[All,1]] (* Harvey P. Dale, May 03 2018 *)

vecextract(A124596, select(t->t==30, A124596[^1]-A124596[^-1],1)) \\ Terms of A124596 with indices of first differences of 30. Gives a(1..230) from A124596(1..10^4). - M. F. Hasler, Jan 02 2020

{ A124596(n) | A124596(n+1) = A124596(n) + 30 }. - M. F. Hasler, Jan 02 2020

A053074 Primes p such that p-24, p and p+24 are consecutive primes.

Original entry on oeis.org

16787, 40063, 42533, 96377, 98597, 104207, 119267, 123887, 160117, 161807, 169283, 181813, 185267, 208553, 209743, 232777, 235723, 243367, 246073, 260363, 261823, 270097, 295387, 295727, 302483, 315223, 331423, 362027, 364103, 373693

Offset: 1

Harvey P. Dale, Feb 25 2000

In other words, balanced primes separated from the next lower and next higher prime neighbors by 24.

			40063 is separated from both the next lower prime and the next higher prime by 24;
104207 - 24 = 104183 is prime, 104207 + 24 = 104231 is prime, and 104207 is the only prime between 104183 and 104231.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A052190.

for i from 1 by 1 to 40000 do if ithprime(i+1) = ithprime(i) +24 and ithprime(i+2) = ithprime(i) + 48 then print(ithprime(i+1)); fi; od; # Zerinvary Lajos, May 04 2007

lst={};Do[p=Prime[n];If[p-Prime[n-1]==Prime[n+1]-p==6*4,AppendTo[lst,p]],{n,2,8!}];lst (* Vladimir Joseph Stephan Orlovsky, May 20 2010 *)
Transpose[Select[Partition[Prime[Range[40000]],3,1],Differences[#]=={24,24}&]][[2]] (* Harvey P. Dale, May 20 2014 *)

a(n) = A052190(n) + 24. - Sean A. Irvine, Dec 06 2021

Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar

Edited by Jon E. Schoenfield, Jan 09 2015

A052377 Primes followed by an [8,4,8]=[d,D-d,d] prime difference pattern of A001223.

Original entry on oeis.org

389, 479, 1559, 3209, 8669, 12269, 12401, 13151, 14411, 14759, 21851, 28859, 31469, 33191, 36551, 39659, 40751, 50321, 54311, 64601, 70229, 77339, 79601, 87671, 99551, 102539, 110261, 114749, 114761, 118661, 129449, 132611, 136511

Offset: 1

Labos Elemer, Mar 22 2000

nonn

A subsequence of A031926. [Corrected by Sean A. Irvine, Nov 07 2021]
a(n)=p, the initial prime of two consecutive 8-twins of primes as follows: [p,p+8] and [p+12,p+12+8], d=8, while the distance of the two 8-twins is 12 (minimal; see A052380(4/2)=12).
Analogous sequences are A047948 for d=2, A052378 for d=4, A052376 for d=10 and A052188-A052199 for d=6k, so that in the [d,D-d,d] difference patterns which follows a(n) the D-d is minimal(=0,2,4; here it is 4).

			p=1559 begins the [1559,1567,1571,1579] prime quadruple consisting of two 8-twins [1559,1567] and[1571,1579] which are in minimal distance, min{D}=1571-1559=12=A052380(8/2).

Cf. A031926, A053325, A052380, A052376, A052378, A052188-A052190.

a(n) is the initial term of a [p, p+8, p+12, p+12+8] quadruple of consecutive primes.

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).

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

A122535 Smallest prime of a triple of successive primes, where the middle one is the arithmetic mean of the other two.

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A052195 Primes p such that p, p+30, p+60 are consecutive primes.

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A053074 Primes p such that p-24, p and p+24 are consecutive primes.

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A052377 Primes followed by an [8,4,8]=[d,D-d,d] prime difference pattern of A001223.

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

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A329578 First of three consecutive primes with common gap 48.

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

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