A094220 -id:A094220 - cp's OEIS frontend

A227284 First primes of arithmetic progressions of 9 primes each with the common difference 210.

199, 409, 3499, 10859, 564973, 1288607, 1302281, 2358841, 3600521, 4047803, 17160749, 20751193, 23241473, 44687567, 50655739, 53235151, 87662609, 100174043, 103468003, 110094161, 180885839, 187874017, 192205147, 221712811, 243051733, 243051943, 304570103

Offset: 1

Sameen Ahmed Khan, Jul 05 2013

nonn

The minimal possible difference in an AP-k is conjectured to be k# for all k > 7.

When a(n+1) = a(n) + 210, as for n = 1, 25, ..., then a(n) is in A094220: start of AP of 10 primes with common distance 210. - M. F. Hasler, Jan 02 2020

			p = 409 then the AP-9 is {409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089} with the difference 9# = 2*3*5*7 = 210.

Sameen Ahmed Khan, Table of n, a(n) for n = 1..47
OEIS wiki, Primes in arithmetic progression.

Cf. A001359, A023241, A023271, A094220, A156204, A227281, A227282, A227283, A227285, A227286.

Clear[p]; d = 210; ap9p = {}; Do[If[PrimeQ[{p, p + d, p + 2*d, p + 3*d, p + 4*d, p + 5*d, p + 6*d, p + 7*d, p + 8*d}] == {True, True, True, True, True, True, True, True, True}, AppendTo[ap9p, p]], {p, 3, 10^9, 2}]; ap9p

v=[1..8]*210; forprime(p=1,,for(i=1,#v,isprime(p+v[i])||next(2));print1(p",")) \\ M. F. Hasler, Jan 02 2020

A227281 First primes of arithmetic progressions of 5 primes each with the common difference 30.

Original entry on oeis.org

7, 11, 37, 107, 137, 151, 277, 359, 389, 401, 541, 557, 571, 877, 1033, 1493, 1663, 2221, 2251, 2879, 3271, 6269, 6673, 6703, 7457, 7487, 9431, 10103, 10133, 10567, 11981, 12457, 12973, 14723, 17047, 19387, 24061, 25643, 25673, 26861, 26891, 27337, 27367

Offset: 1

Sameen Ahmed Khan, Jul 05 2013

nonn

The minimal possible difference in an AP-k is conjectured to be k# for all k > 7.

For k = 5, we have d = 3# = 6 and there is ONLY one AP-5 with this difference: {5, 11, 17, 23, 29}.

			p = 11 then {11, 11 + 1*30, 11 + 2*30, 11 + 3*30, 11 + 4*30} = {11, 41, 71, 101, 131}, which is 5 primes in arithmetic progression with the difference 5# = 30.

Sameen Ahmed Khan, Table of n, a(n) for n = 1..1184

Cf. A001359, A023241, A023271, A094220, A156204, A227282, A227283, A227284, A227285, A227286.

Clear[p]; d = 30; ap5p = {}; Do[If[PrimeQ[{p, p + d, p + 2*d, p + 3*d, p + 4*d}] == {True, True, True, True, True}, AppendTo[ap5p, p]], {p, 3, 25000, 2}]; ap5p

A227282 First primes of arithmetic progressions of 7 primes each with the common difference 210.

Original entry on oeis.org

47, 179, 199, 409, 619, 829, 881, 1091, 1453, 3499, 3709, 3919, 10529, 10627, 10837, 10859, 11069, 11279, 14423, 20771, 22697, 30097, 30307, 31583, 31793, 32363, 41669, 75703, 93281, 95747, 120661, 120737, 120871, 120947, 129287, 140603, 153319, 153529

Offset: 1

Sameen Ahmed Khan, Jul 05 2013

nonn

The minimal possible difference in an AP-k is conjectured to be k# for all k > 7.

For k = 7, we have d = 5*5# = 150 and there is ONLY one AP-7 with this difference: {7, 157, 307, 457, 607, 757, 907}.

			p = 179 then the AP-5 is {179, 389, 599, 809, 1019, 1229, 1439} with the difference 7# = 210.

Sameen Ahmed Khan and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 2484 terms from Khan)

Cf. A001359, A023241, A023271, A094220, A156204, A227281, A227283, A227284, A227285, A227286.

Clear[p]; d = 210; ap7p = {}; Do[If[PrimeQ[{p, p + d, p + 2*d, p + 3*d, p + 4*d, p + 5*d, p + 6*d}] == {True, True, True, True, True, True, True}, AppendTo[ap7p, p]], {p, 3, 10^9, 2}]; ap7p
Select[Prime[Range[15000]],And@@PrimeQ[NestList[210+#&,#,6]]&] (* Harvey P. Dale, Nov 16 2013 *)

is(p)=forstep(k=p,p+1260,210,if(!isprime(k),return(0)));1 \\ Charles R Greathouse IV, Dec 19 2013

A227283 First primes of arithmetic progressions of 8 primes each with the common difference 210.

Original entry on oeis.org

199, 409, 619, 881, 3499, 3709, 10627, 10859, 11069, 30097, 31583, 120661, 120737, 153319, 182537, 471089, 487391, 564973, 565183, 825991, 1010747, 1280623, 1288607, 1288817, 1302281, 1302491, 1395209, 1982599, 2358841, 2359051, 2439571, 3161017, 3600521

Offset: 1

Sameen Ahmed Khan, Jul 05 2013

nonn

The minimal possible difference in an AP-k is conjectured to be k# for all k > 7.

Sameen Ahmed Khan, Table of n, a(n) for n = 1..150

Cf. A001359, A023241, A023271, A094220, A156204, A227281, A227282, A227284, A227285, A227286.

Clear[p]; d = 210; ap8p = {}; Do[If[PrimeQ[{p, p + d, p + 2*d, p + 3*d, p + 4*d, p + 5*d, p + 6*d, p + 7*d}] == {True, True, True, True, True, True, True, True}, AppendTo[ap8p, p]], {p, 3, 3000000, 2}]; ap8p
Select[Prime[Range[260000]],AllTrue[NestList[#+210&,#,7],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 03 2018 *)

A227285 First primes of arithmetic progressions of 11 primes each with the common difference 2310.

Original entry on oeis.org

60858179, 186874511, 291297353, 1445838451, 2943023729, 4597225889, 7024895393, 8620560607, 8656181357, 19033631401, 20711172773, 25366690189, 27187846201, 32022299977, 34351919351

Offset: 1

Sameen Ahmed Khan, Jul 05 2013

nonn

The minimal possible difference in an AP-k is conjectured to be k# for all k > 7.

16th term is greater than 40*10^9.

			p = 186874511 then the AP-11 is {186874511, 186876821, 186879131, 186881441, 186883751, 186886061, 186888371, 186890681, 186892991, 186895301, 186897611} with the difference 11# = 2*3*5*7*11 = 2310.

Zak Seidov, Table of n, a(n) for n = 1..21

Cf. A001359, A023241, A023271, A094220, A156204, A227281, A227282, A227283, A227284, A227286.

Clear[p]; d = 2310; ap11p = {}; Do[If[PrimeQ[{p, p + d, p + 2*d, p + 3*d, p + 4*d, p + 5*d, p + 6*d, p + 7*d, p + 8*d, p + 9*d, p + 10*d}] == {True, True, True, True, True, True, True, True, True, True, True}, AppendTo[ap11p, p]], {p, 3, 40*10^9, 2}]; ap11p
ap11Q[n_]:=AllTrue[Rest[NestList[2310+#&,n,10]],PrimeQ]; Select[Prime[ Range[ 148*10^7]],ap11Q] (* The program uses the AllTrue function from Mathematica version 10 *) (* The program will take a long time to run *) (* Harvey P. Dale, Oct 27 2019 *)

a(16)-a(21) from Zak Seidov, Jul 07 2014

A227286 First primes of arithmetic progressions of 13 primes each with the common difference 30030.

Original entry on oeis.org

14933623, 2085471361, 132420258931, 185041386139, 682539280751, 834172298383, 834172328413, 856378247603, 856378277633, 888867525577, 931115864233, 1059709587163, 1345030977911, 1360910561113, 1578280523803, 1973348047529, 1988253536611, 2083502941613

Offset: 1

Sameen Ahmed Khan, Jul 05 2013

nonn

The minimal possible difference in an arithmetic progression of k primes is conjectured to be k# = A034386(k) for all k > 7. 13# = 30030.

			p = 2085471361 then the AP-13 is {2085471361, 2085501391, 2085531421, 2085561451, 2085591481, 2085621511, 2085651541, 2085681571, 2085711601, 2085741631, 2085771661, 2085801691, 2085831721} with the difference 13# = 2*3*5*7*11*13 = 30030.

Cf. A001359, A023241, A023271, A094220, A156204, A227281, A227282, A227283, A227284, A227285.

Clear[p]; d = 30030; ap13p = {}; Do[If[PrimeQ[{p, p + d, p + 2*d, p + 3*d, p + 4*d, p + 5*d, p + 6*d, p + 7*d, p + 8*d, p + 9*d, p + 10*d, p + 11*d, p + 12*d}] == {True, True, True, True, True, True, True, True, True, True, True, True, True}, AppendTo[ap13p, p]], {p, 3, 41*10^9, 2}]; ap13p

More terms from Jens Kruse Andersen, Jun 27 2014

A033168 Longest arithmetic progression of primes with difference 210 and minimal initial term.

Original entry on oeis.org

199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089

Offset: 0

Manuel Valdivia, Apr 22 1998

Since 210 == 1 (mod 11), a progression of primes with difference 210 can't have more than ten terms because there is exactly one multiple of 11 within each run of eleven consecutive terms. For example, 2089 + 210 = 2299 = 11^2 * 19. - Alonso del Arte, Dec 22 2017, edited by M. F. Hasler, Jan 02 2020

After 199, the next prime which starts a CPAP-10 with common gap 210 is 243051733. See A094220 for further starting points. - M. F. Hasler, Jan 02 2020

Paul Glendinning, Math in Minutes: 200 Key Concepts Explained in an Instant. New York, London: Quercus (2013): pp. 316-317.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 143.

Tanya Khovanova, Non Recursions.
OEIS wiki, Primes in arithmetic progression.
Index entries for sequences related to primes in arithmetic progressions.

Row 10 of A086786, A113470, A133276, A133277.

Cf. A094220.

199 + 210*Range[0, 9] (* Paolo Xausa, Sep 14 2024 *)

forprime(p=1,,for(i=1,9,isprime(p+i*210)||next(2)); return([p+d|d<-[0..9]*210])) \\ M. F. Hasler, Jan 02 2020

a(n) = a(0) + n*210 for 0 <= n <= 9. - M. F. Hasler, Jan 02 2020

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A227284 First primes of arithmetic progressions of 9 primes each with the common difference 210.

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A227281 First primes of arithmetic progressions of 5 primes each with the common difference 30.

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A227282 First primes of arithmetic progressions of 7 primes each with the common difference 210.

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A227283 First primes of arithmetic progressions of 8 primes each with the common difference 210.

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A227285 First primes of arithmetic progressions of 11 primes each with the common difference 2310.

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A227286 First primes of arithmetic progressions of 13 primes each with the common difference 30030.

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A033168 Longest arithmetic progression of primes with difference 210 and minimal initial term.

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