A052195 -id:A052195 - cp's OEIS frontend

A089218 Erroneous version of A052195.

69593, 110651, 228647, 250889, 318919, 396449, 421913, 498271, 507431, 554317

Offset: 1

dead

A052243 Initial prime in set of (at least) 4 consecutive primes in arithmetic progression with difference 30.

642427, 1058861, 3431903, 4176587, 4560121, 4721047, 5072269, 5145403, 5669099, 5893141, 6248969, 6285047, 6503179, 6682969, 8545357, 8776121, 8778739, 9490571, 9836227, 9843019, 9843049, 10023787, 11697979, 12057919, 12340313, 12687119, 12794641, 12845849

Offset: 1

Labos Elemer, Jan 31 2000

nonn

Primes p such that p, p+30, p+60, p+90 are consecutive primes.

The analogous sequence for a CPAP-5 (at least five consecutive primes in arithmetic progression) with gap 30 does not have its own entry in the OEIS, but for over 500 terms it is identical to A059044. The CPAP-6 analog is A058362. - M. F. Hasler, Jan 02 2020

			642427, 642457, 642487, 642517 are consecutive primes, so 642427 is in the sequence.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms n = 1..200 from M. F. Hasler)
OEIS wiki, Consecutive primes in arithmetic progression: CPAP with given gap, updated Jan. 2020.
Index entries for sequences related to primes in arithmetic progressions

Analogous sequences (start of CPAP-4 with common difference in square brackets): A033451 [6], A033447 [12], A033448 [18], A052242 [24], A052243 [this: 30], A058252 [36], A058323 [42], A067388 [48], A259224 [54], A210683 [60].
Subsequence of A052195 and of A054800 (start of CPAP-4 with any common difference).
See also A059044 (start of CPAP-5), A058362 (CPAP-6).

f:=func; a:=[]; for p in PrimesInInterval(2,13000000) do if  (f(p)-p eq 30) and (f(f(p))-p eq 60) and (f(f(f(p)))-p eq 90) then Append(~a,p); end if; end for; a; // Marius A. Burtea, Jan 04 2020

p := 2 : q := 3 : r := 5 : s := 7 : for i from 1 do if q-p = 30 and r-q=30 and s-r=30 then printf("%d,\n",p) ; fi ; p := q ; q := r ; r := s ; s := nextprime(r) ; od: # R. J. Mathar, Apr 12 2008

p=2; q=3; r=5; s=7; A052243 = Reap[For[i=1, i<1000000, i++, If[ q-p == 30 && r-q == 30 && s-r == 30 , Print[p]; Sow[p]]; p=q; q=r; r=s; s=NextPrime[r]]][[2, 1]] (* Jean-François Alcover, Jun 28 2012, after R. J. Mathar *)
Transpose[Select[Partition[Prime[Range[1100000]],4,1],Union[ Differences[#]] =={30}&]][[1]] (* Harvey P. Dale, Jun 17 2014 *)

A052243(n,p=2,print_all=0,g=30,c,o)={forprime(q=p+1,,if(p+g!=p=q, next, q!=o+2*g, c=2, c++>3, print_all&& print1(o-g","); n--||break); o=q-g);o-g} \\ optional 2nd arg specifies starting point, allows to define:
next_A052243(p)=A052243(1,p+1) \\ replacing older code from 2008. - M. F. Hasler, Oct 26 2018

A052243 = { A052195(n) | A052195(n+1) = A052195(n) + 30 }. - M. F. Hasler, Jan 02 2020

More terms from Harvey P. Dale, Nov 19 2000

Edited by N. J. A. Sloane, Apr 28 2008, at the suggestion of R. J. Mathar

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

Original entry on oeis.org

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

A052239 Smallest prime p in set of 4 consecutive primes in arithmetic progression with common difference 6n.

Original entry on oeis.org

251, 111497, 74453, 1397609, 642427, 5321191, 23921257, 55410683, 400948369, 253444777, 1140813701, 491525857, 998051413, 2060959049, 4480114337, 55140921491, 38415872947, 315392068463, 15162919459, 60600021611, 278300877401, 477836574947, 1486135570643

Offset: 1

Labos Elemer, Jan 31 2000

See also the less restrictive A054701 where the gaps are multiples 6n. - M. F. Hasler, Nov 06 2018

			a(5) = 642427, 642457, 642487, 642517 are the smallest consecutive primes with 3 consecutive gaps of 30, cf. A052243.
From _M. F. Hasler_, Nov 06 2018: (Start)
Other terms are also initial terms of corresponding sequences:
a(1) = 251 = A033451(1) = A054800(1), start of first CPAP-4 with common gap of 6,
a(2) = 111497 = A033447(1), start of first CPAP-4 with common gap of 12,
a(3) = 74453 = A033448(1) = A054800(25), first CPAP-4 with common gap of 18,
a(4) = 1397609 = A052242(1), start of first CPAP-4 with common gap of 24,
a(5) = 642427 = A052243(1) = A052195(16), first CPAP-4 with common gap of 30,
a(6) = 5321191 = A058252(1) = A161534(26), first CPAP-4 with common gap 36 = 6^2,
a(7) = 23921257 = A058323(1), start of first CPAP-4 with common gap of 42,
a(8) = 55410683 = A067388(1), start of first CPAP-4 with common gap of 48,
a(9) = 400948369 = A259224(1), start of first CPAP-4 with common gap of 54,
a(10) = 253444777 = A210683(1) = A089234(417), CPAP-4 with common gap of 60,
a(11) = 1140813701 = A287547(1), start of first CPAP-4 with common gap of 66,
a(12) = 491525857 = A287550(1), start of first CPAP-4 with common gap of 72,
a(13) = 998051413 = A287171(1), start of first CPAP-4 with common gap of 78,
a(14) = 2060959049 = A287593(1), start of first CPAP-4 with common gap of 84,
a(15) = 4480114337 = A286817(1) = A204852(444), common distance 90. (End)

Jerry M Lagrou, Table of n, a(n) for n = 1..42
Index entries for sequences related to primes in arithmetic progressions

Initial terms of sequences A033451, A033447, A033448, A052242, A052243, A058252, A058323, A067388, A259224, A210683.
Range is a subset of A054800: start of 4 consecutive primes in arithmetic progression (CPAP-4).
Cf. A054701: gaps are possibly distinct multiples of 6n (not CPAP's).

Transpose[Flatten[Table[Select[Partition[Prime[Range[2000000]],4,1], Union[ Differences[ #]] =={6n}&,1],{n,7}],1]][[1]] (* Harvey P. Dale, Aug 12 2012 *)

a(n, p=[2, 0, 0], d=6*[n, n, n])={while(p+d!=p=[nextprime(p[1]+1), p[1], p[2]], ); p[3]-d[3]} \\ after M. F. Hasler in A052243; Graziano Aglietti (mg5055(AT)mclink.it), Aug 22 2010, Corrected by M. F. Hasler, Nov 06 2018

A052239(n, p=2, c, o)={n*=6; forprime(q=p+1, , if(p+n!=p=q, next, q!=o+2*n, c=2, c++>3, break); o=q-n); o-n} \\ M. F. Hasler, Nov 06 2018

More terms from Labos Elemer, Jan 04 2002
a(7) corrected and more terms added by Graziano Aglietti (mg5055(AT)mclink.it), Aug 22 2010
a(15)-a(20) from Donovan Johnson, Oct 05 2010
a(21)-a(23) from Donovan Johnson, May 23 2011

A124596 Primes p such that q-p = 30, where q is the next prime after p.

Original entry on oeis.org

4297, 4831, 5351, 5749, 6491, 6917, 7253, 7759, 7963, 8389, 8893, 13063, 13187, 13933, 13967, 14251, 14983, 16381, 16573, 17627, 18553, 18869, 20563, 21283, 21347, 21617, 23633, 23689, 24251, 25189, 26053, 26597, 27299, 27367, 27551, 28319, 28979, 29537

Offset: 1

N. J. A. Sloane, Dec 19 2006

nonn

These are the primes p = prime(k) where A001223(k) = 30. - N. J. A. Sloane, Dec 25 2019

Primes p such that r-q=q-p=30, where p, q, r are three successive primes, are A052195 = (69593, 110651, 134609, 228647, 237791, 250889, 303157, 24907,...). Primes p such that s-r=r-q=q-p=30, where p, q, r, s are four successive primes, are A052243 = (642427, 1058861, 3431903, 4176587, 4560121, 4721047, 5072269, 5145403, ...). - Zak Seidov, Dec 20 2006, edited by M. F. Hasler, Apr 04 2013

Remi Eismann, Table of n, a(n) for n = 1..10000
K. Soundararajan, Small gaps between prime numbers: The work of Goldston-Pintz-Yildirim, Bull. Amer. Math. Soc., 44 (2007), 1-18. DOI: 10.1090/S0273-0979-06-01142-6
K. Soundararajan, Small gaps between prime numbers: The work of Goldston-Pintz-Yildirim, arXiv:math/0605696 [math.NT], 2006.
Index entries for primes, gaps between

Cf. A001223, A052195, A052243.

See also A052187 and references therein.

max = 30000;
Reap[For[p = 2; q = 3, p < max, p = q, q = NextPrime[p]; If[q - p == 30, Sow[p]]]][[2, 1]] (* Jean-François Alcover, Sep 02 2018 *)
Select[Partition[Prime[Range[5000]],2,1],#[[2]]-#[[1]]==30&][[All,1]] (* Harvey P. Dale, Dec 25 2019 *)

A052197 Primes p such that p, p+36, p+72 are consecutive primes.

Original entry on oeis.org

255767, 704321, 806821, 884501, 913067, 1065137, 1216177, 1448497, 1526191, 1532471, 1640971, 1918571, 2071087, 2275067, 2276431, 2336671, 2347591, 2376721, 2778547, 3098561, 3190601, 3248941, 3259001, 3452107, 3558481

Offset: 1

Labos Elemer, Jan 28 2000

nonn

Old name was: Primes p(k) such that p(k+2)-p(k+1)=p(k+1)-p(k)=36.

			a(3) = 704321 is followed by 704357 and 704393, consecutive primes with equal distance of d = 36.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..3000 from Zak Seidov)

Subsequence of A134117.

Cf. A001223, A033451, A047948, A052195.

Select[Partition[Prime[Range[255000]],3,1],Differences[#]=={36,36}&][[All,1]] (* Harvey P. Dale, Feb 16 2018 *)

is(n)=nextprime(n+1)==n+36 && nextprime(n+37)==n+72 && isprime(n) \\ Charles R Greathouse IV, Jan 07 2013

New name from Charles R Greathouse IV, Jan 07 2013

A052187 a(n) is the smallest prime p such that p, p+d, and p+2d are consecutive primes where d = 2 for n = 1 and d = 6*(n-1) for n > 1.

Original entry on oeis.org

3, 47, 199, 20183, 16763, 69593, 255767, 247099, 3565931, 6314393, 4911251, 12012677, 23346737, 43607351, 34346203, 36598517, 51041957, 460475467, 652576321, 742585183, 530324329, 807620651, 2988119207, 12447231761, 383204539, 4470607951, 5007182707

Offset: 1

Labos Elemer, Jan 28 2000

nonn

The first term 3 is anomalous since for all others d is divisible by 6. These are minimal terms if in A047948 d=6 is replaced by possible differences: (2), 6, 12, 18, ..., 54, 60.

a(54) > 5*10^13, while a(55) = 46186474937633. - Giovanni Resta, Apr 08 2013

			a(2)=47 and it is the lower border of a dd pattern: 47[6 ]53[6 ]59. a(10)=6314393 and a(10)+54=6314447, a(10)+108=6314501 are consecutive primes and 6314393 is the smallest prime prior to a (54,54) difference pattern of A001223.

Jerry M Lagrou, Table of n, a(n) for n = 1..72 (terms 1..39 from Donovan Johnson, terms 40..53 from Giovanni Resta)

Cf. A001223, A047948, A052160, A054342.

Cf. A052188, A052189, A052195, A052196, A052197, A052198.

a = Table[0, {100}]; NextPrime[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = q = r = 0; Do[r = NextPrime[r]; If[r + p == 2q && r - q < 201 && a[[(r - q)/2]] == 0, a[[(r - q)/2]] = p]; p = q; q = r, {n, 1, 10^6}]; a (* Typos fixed by Zak Seidov, May 01 2020 *)

list(n)=ve=vector(n);ppp=2;pp=3;forprime(p=5,,d=p-pp;if(pp-ppp==d,i=d\6+1;if(i<=n&&ve[i]==0,ve[i]=ppp;print1(".");vecprod(ve)>0&&return(ve)));ppp=pp;pp=p) \\ Jeppe Stig Nielsen, Apr 17 2022

The least prime(k) such that prime(k+1) = (prime(k) + prime(k+2))/2 and prime(k+1) - prime(k) = d is either 2 or divisible by 6.

a(1) = A054342(1) - 2. For n>1, a(n) = A054342(n) - 6*(n-1). - Jeppe Stig Nielsen, Apr 16 2022

More terms from Labos Elemer, Jan 04 2002
More terms from Robert G. Wilson v, Jan 06 2002
Definition clarified by Harvey P. Dale, Aug 29 2012
a(23)-a(27) from Donovan Johnson, Aug 30 2012
Name edited by Jon E. Schoenfield, Nov 30 2023

A053075 Primes p such that p-30, p, p+30 are consecutive primes.

Original entry on oeis.org

69623, 110681, 134639, 228677, 237821, 250919, 303187, 318949, 396479, 421943, 498301, 507461, 535273, 554347, 629653, 642457, 642487, 668273, 692191, 716033, 729821, 780553, 782611, 790927, 801247, 825161, 829319, 847423, 892321, 902903

Offset: 1

Harvey P. Dale, Feb 25 2000

Original name: Primes p(k) such that p(k) - p(k-1) = p(k+1) - p(k) = 30.

			110681 is separated from both the next lower prime and the next higher prime by 30

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

Cf. A052195 (a(n)-30: first of the triplets) and cross-references there.

Subsequence of A124596 (primes followed by gap 30).

lst={}; Do[p=Prime[n]; If[p-Prime[n-1] == Prime[n+1]-p == 6*5, AppendTo[lst,p]], {n,2,2*8!}]; lst (* Vladimir Joseph Stephan Orlovsky, May 20 2010 *)

is_A053075(n)={precprime(n-1)==n-30&&nextprime(n+1)==n+30&&isprime(n)} \\ M. F. Hasler, Jan 02 2020

a(n) = A052195(n) + 30. - Zak Seidov, Dec 21 2012

A052195 = { A124596(n) | A124596(n-1) = A124596(n) - 30 }. - M. F. Hasler, Jan 02 2020

Name edited to conform with style sheet and A052195 etc. - M. F. Hasler, Jan 02 2020

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).

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A089218 Erroneous version of A052195.

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A052243 Initial prime in set of (at least) 4 consecutive primes in arithmetic progression with difference 30.

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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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A052239 Smallest prime p in set of 4 consecutive primes in arithmetic progression with common difference 6n.

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A124596 Primes p such that q-p = 30, where q is the next prime after p.

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A052197 Primes p such that p, p+36, p+72 are consecutive primes.

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A052187 a(n) is the smallest prime p such that p, p+d, and p+2d are consecutive primes where d = 2 for n = 1 and d = 6*(n-1) for n > 1.

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