A023241 -id:A023241 - cp's OEIS frontend

A046140 Duplicate of A023241.

5, 7, 11, 17, 31, 41, 47, 61, 67, 97, 101, 151, 167, 227, 251, 257, 271, 347, 367, 557, 587

Offset: 1

dead

A358572 Smallest prime p in a sexy prime triple such that (p-3)/2 is also the smallest prime in a sexy prime triple (A023241).

17, 97, 1117, 1217, 2897, 130337, 188857, 207997, 221197, 324517, 610817, 900577, 1090877, 1452317, 1719857, 1785097, 2902477, 3069917, 3246317, 4095097, 4536517, 4977097, 5153537, 5517637, 5745557, 6399677, 7168277, 7351957, 7588697, 7661077, 8651537, 8828497, 9153337

Offset: 1

Lamine Ngom, Nov 23 2022

nonn

Also numbers m such that m-4, m-1, m+5, m+8, m+11 and m+20 cannot be represented as x*y + x + y, with x >= y > 1 (A254636).
Subsequence of A358571.
Number of terms < 10^k: 0, 2, 2, 5, 5, 12, 34, 150, 655, ...
All terms p and (p-3)/2 have a final decimal digit of 7. This follows from considering possibilities modulo 10 and implies p + 18 and (p-3)/2 + 18 are divisible by 5 and hence composite. Both p and (p-3)/2 are therefore also terms of A046118. - Andrew Howroyd, Dec 31 2022

			97 is the smallest prime in the sexy prime triple (97, 103, 109), and the triple (47 = (97 - 3)/2, 47 + 6, 47 + 12) forms another sexy prime triple. Hence 97 is in the sequence.

Cf. A023201, A023241, A046118, A255361, A254636, A256386, A358571.

Select[Prime[Range[700000]], AllTrue[Join[# + {6,12}, (#-3)/2 + {0, 6, 12}], PrimeQ] &] (* Amiram Eldar, Nov 23 2022 *)

istriple(p)={isprime(p) && isprime(p+6) && isprime(p+12)}
isok(p)={istriple(p) && istriple((p-3)/2)}
{ forprime(p=1,10^7,if(isok(p), print1(p, ", "))) } \\ Andrew Howroyd, Dec 30 2022

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

Original entry on oeis.org

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

A279765 Primes p such that p+24 and p+48 are also primes.

Original entry on oeis.org

5, 13, 19, 23, 59, 79, 83, 89, 103, 149, 233, 269, 283, 349, 373, 409, 419, 439, 443, 499, 523, 569, 593, 653, 709, 773, 829, 839, 859, 863, 929, 1039, 1069, 1259, 1279, 1399, 1423, 1559, 1699, 1753, 1823, 1949, 1979, 2039, 2063, 2089, 2113, 2309, 2333, 2393

Offset: 1

Gerhard Kirchner, Dec 18 2016

nonn

Subsequence of A033560. The triples have the form (p,p+d,p+2d). The current sequence (d=24) continues A023241 (d=6), A185022 (d=12) and A156109 (d=18). The frequencies of such triples and the triple (p, p+3±1, p+6) in A007529 do not differ very much (see table in the link "comparison of triples"). For creating the b-file I used a file of prime differences, divided by 2 (extension of A028334). For filling the table I analyzed primes up to 10^9.

Annotation: The algorithm using a file of primes or prime differences is not difficult but not as easy as using a function like isprime(n). On the other hand, such a function needs computing time which is not negligible for large numbers.

			First term: 5, 5 + 24 = 29 and 5 + 48 = 53 are all primes.

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

Cf. A023241, A033560, A156109, A185022.

Select[Prime@Range@500, PrimeQ[# + 24] && PrimeQ[# + 48] &] (* Robert G. Wilson v, Dec 18 2016 *)

is(n) = for(k=0, 2, if(!ispseudoprime(n+24*k), return(0))); 1 \\ Felix Fröhlich, Dec 26 2016

A377317 Numbers k such that prime(k), prime(k)+6, and prime(k)+12 are primes.

Original entry on oeis.org

3, 4, 5, 7, 11, 13, 15, 18, 19, 25, 26, 36, 39, 49, 54, 55, 58, 69, 73, 102, 107, 110, 111, 116, 118, 129, 160, 164, 182, 184, 187, 194, 199, 206, 210, 218, 225, 229, 234, 236, 252, 253, 260, 271, 272, 275, 284, 285, 291, 300, 321, 339, 351, 352, 358, 387, 388

Offset: 1

Kritsada Moomuang, Oct 24 2024

nonn

			5 is in this sequence because: prime(5) = 11, and 11+6 = 17 and 11+12 = 23 are primes.

Cf. A000040, A000720, A023241.

Supersequence of A377318.

Select[Range[1, PrimePi[3000]], PrimeQ[Prime[#] + 6] && PrimeQ[Prime[#] + 12] &]

for(k=1, primepi(3000), p = prime(k); if(isprime(p+6) && isprime(p+12), print(k)))

a(n) = pi(A023241(n)).

cp's OEIS Frontend

A046140 Duplicate of A023241.

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A358572 Smallest prime p in a sexy prime triple such that (p-3)/2 is also the smallest prime in a sexy prime triple (A023241).

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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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A279765 Primes p such that p+24 and p+48 are also primes.

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