A033447 -id:A033447 - cp's OEIS frontend

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

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

A210683 Primes p such that p, p+60, p+120, p+180 are consecutive primes.

Original entry on oeis.org

253444777, 271386581, 286000489, 415893013, 475992773, 523294549, 620164949, 794689481, 838188877, 840725323, 846389227, 884106599, 884951807, 908725507, 941796223, 952288331, 971614151, 1002290693, 1003166771, 1006976797, 1053792359, 1097338313, 1163141201

Offset: 1

Zak Seidov, May 09 2012

nonn

Subsequence of A089234 which itself is a subsequence of A126771:
a(1) = 253444777 = A089234(417) = A126771(81526),
a(36) = 1998782563 = A089234(5579) = A126771(788920).

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

Cf. A089234, A126771.
Analogous sequences (start of CPAP-4, with common difference in square brackets): A033451 [6], A033447 [12], A033448 [18], A052242 [24], A052243 [30], A058252 [36], A058323 [42], A067388 [48], A259224 [54].
Cf. A054800: union of all sequences of this type (start of CPAP-4).

A210683(n, p=2, v=1, g=60, c, o)={forprime(q=p+1, , if(p+g!=p=q, next, q!=o+2*g, c=2, c++>3, v&& print1(o-g", "); n--||break); o=q-g); o-g} \\ Can be used as next(p)=A210683(1, p) to get the next term, e.g.:
p=0; A210683_vec=vector(10,i,p=A210683(1,p)) \\ Will take a long time! - M. F. Hasler, Oct 26 2018

A259224 Initial primes in sets of 4 consecutive primes with common gap 54.

Original entry on oeis.org

400948369, 473838319, 583946599, 678953059, 816604199, 972598819, 1136526949, 1466715139, 1475790529, 1499794999, 1502149559, 1610895679, 1643313869, 1673057219, 1686181579, 1845792019, 1867046639, 1907478889, 1992202439, 2011077869, 2030490479, 2207714969

Offset: 1

Zak Seidov, Jun 21 2015

nonn

All terms are == {19,29} mod 30.

Lars Blomberg, Table of n, a(n) for n = 1..10000 (The first 102 terms from Zak Seidov)

Start of CPAP-4 with given common difference (in square brackets): A033451 [6], A033447 [12], A033448 [18], A052242 [24], A052243 [30], A058252 [36], A058323 [42], A067388 [48], A259224 [this: 54], A210683 [60].

Subsequence of A054800: start of a CPAP-4 with arbitrary common difference.

A259224(n, p=2, v=1, g=54, c, o)={forprime(q=p+1, , if(p+g!=p=q, next, q!=o+2*g, c=2, c++>3, v&& print1(o-g", "); n--||break); o=q-g); o-g} \\ Can be used as next(p)=A259224(1,p+1) to get the next term, e.g.:
p=0; A259224_vec=vector(10,i,p=A259224(1,p+1)) \\ Will be slow! - M. F. Hasler, Oct 26 2018

A058362 Initial primes of sets of 6 consecutive primes in arithmetic progression.

Original entry on oeis.org

121174811, 1128318991, 2201579179, 2715239543, 2840465567, 3510848161, 3688067693, 3893783651, 5089850089, 5825680093, 6649068043, 6778294049, 7064865859, 7912975891, 8099786711, 9010802341, 9327115723, 9491161423, 9544001791, 10101930253, 10523406343, 13193702321

Offset: 1

Harvey Dubner (harvey(AT)dubner.com), Dec 18 2000

nonn

For all the terms listed so far, the common difference is equal to 30. These are the smallest such sets.
It is conjectured that there exist arbitrarily long sequences of consecutive primes in arithmetic progression. As of December 2000 the record is 10 primes.
All terms are congruent to 9 (mod 14). - Zak Seidov, May 03 2017
The first CPAP-6 with common difference 60 starts at 293826343073 ~ 3*10^11, cf. A210727. [With a slope of a(n)/n ~ 5*10^8 this would correspond to n ~ 600.] This sequence consists of first members of pairs of consecutive primes in A059044. Conversely, a pair of consecutive primes in this sequence starts a CPAP-7. This must have a common difference >= 210. As of today, the smallest known CPAP-7 starts at 382003672700092872707633 ~ 3.8*10^23, cf. Andersen link. - M. F. Hasler, Oct 27 2018
The common difference of 60 first occurs at a larger-than-expected prime. The first CPAP-6 with common difference 90 starts at 8560443932347. The first CPAP-6 with common difference 120 starts at 1925601119017087. - Jerry M Lagrou, Jan 01 2024

Zak Seidov, Table of n, a(n) for n = 1..102
Jens K. Andersen, The Largest Known CPAP's, updated Sep 2018.
OEIS Wiki, Consecutive primes in arithmetic progression, updated Jan 2020.
Index entries for sequences related to primes in arithmetic progressions

Cf. A006560: first prime to start a CPAP-n.
Cf. A033451, A033447, A033448, A052242, A052243, A058252, A058323, A067388: start of CPAP-4 with common difference 6, 12, 18, ..., 48.
Cf. A054800: start of 4 consecutive primes in arithmetic progression (CPAP-4).
Cf. A052239: starting prime of first CPAP-4 with common difference 6n.
Cf. A059044: starting primes of CPAP-5.
Cf. A210727: starting primes of CPAP-5 with common difference 60.

p=c=g=P=0;forprime(q=1,, p+g==(p+=g=q-p)|| next; q==P+2*g&& c++|| c=3; c>5&& print1(P-3*g,","); P=q-g) \\ M. F. Hasler, Oct 26 2018

Equals { A059044(i) | A059044(i+1) = A151800(A059044(i)) }, A151800 = nextprime. - M. F. Hasler, Oct 30 2018

Corrected by Jud McCranie, Jan 04 2001
a(11)-a(18) from Donovan Johnson, Sep 05 2008
Comment split off from Name (to clarify definition) by M. F. Hasler, Oct 27 2018

A253140 Smallest of three consecutive primes in arithmetic progression with common difference 24 and digit sum prime.

Original entry on oeis.org

89, 373, 773, 863, 1279, 2063, 2089, 2399, 2663, 2753, 3299, 4153, 4373, 5879, 6173, 6263, 6779, 7079, 7499, 7853, 9473, 10453, 11399, 12253, 12479, 14699, 16763, 19379, 21163, 21563, 25073, 29363, 32189, 33599, 40063, 41879, 42773, 50053, 50363, 52673, 56453

Offset: 1

K. D. Bajpai, Dec 27 2014

			a(1) = 89: 89 + 24 = 113; 113 + 24 = 137; all three are prime. Their digit sums 8+9 = 17, 1+1+3 = 5 and 1+3+7 = 11 are also prime.
a(2) = 373: 373 + 24 = 397; 397 + 24 = 421; all three are prime. Their digit sums 3+7+3 = 13, 3+9+7 = 19 and 4+2+1 = 7 are also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000040, A033447, A062088.

A253140 = {}; Do[d = 24; k = Prime[n]; k1 = k+d; k2 = k+2d; If[PrimeQ[k1] && PrimeQ[k2] && PrimeQ[Plus@@IntegerDigits[k]] && PrimeQ[Plus@@IntegerDigits[k1]] && PrimeQ[Plus@@IntegerDigits[k2]], AppendTo[A253140,k]], {n,20000}]; A253140
tcpQ[n_]:=Module[{a=n+24,b=n+48},AllTrue[{a,b},PrimeQ]&&AllTrue[Total/@ (IntegerDigits/@{n,a,b}),PrimeQ]]; Select[Prime[Range[6000]],tcpQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 16 2016 *)

A248085 Initial prime of 4 primes in arithmetic progression with difference 12.

Original entry on oeis.org

5, 7, 17, 47, 127, 227, 257, 397, 467, 607, 997, 1447, 1487, 1697, 1877, 2647, 3307, 3547, 3907, 4217, 4987, 5407, 6287, 6947, 7297, 7537, 7817, 10067, 10627, 11047, 11777, 12227, 12577, 13147, 14747, 15137, 15737, 15877, 17827, 19727, 19937, 20707, 21577, 22027, 22247, 23017, 24097, 26017

Offset: 1

Zak Seidov, Oct 01 2014

nonn

Or, primes p such that p + 12, p + 24 and p + 36 are also primes.
Primes are not necessarily consecutive ones. A033447 is subsequence: a(92) = 111497 = A033447(1), a(144) = 258527 = A033447(2), etc.
The only case with p + 48 prime is p = 5, in all other cases p + 48 is divisible by 5.
All terms >5 are congruent to 7 (mod 10). - Zak Seidov, Jun 12 2018

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A033447.

A248085:=n->`if`(isprime(n) and isprime(n+12) and isprime(n+24) and isprime(n+36), n, NULL): seq(A248085(n), n=1..10^5); # Wesley Ivan Hurt, Oct 01 2014

Select[Prime[Range[1000]], PrimeQ[# + 12] && PrimeQ[# + 24] && PrimeQ[# + 36] &] (* Alonso del Arte, Oct 01 2014 *)
Select[Prime[Range[3000]],AllTrue[#+{12,24,36},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 08 2016 *)

forprime(p=5,10^5,isprime(p+12)&&isprime(p+24)&&isprime(p+36)&&print1(p","))

A287547 Initial prime in set of 4 consecutive primes in arithmetic progression with difference 66.

Original entry on oeis.org

1140813701, 1314331181, 1729804331, 2615969891, 2765625631, 3827771821, 4266876641, 4348917061, 4700742041, 4845745831, 4877408441, 5311420901, 5395463741, 5409482081, 5693097391, 5816498981, 5902417331, 6173160871, 6692523011, 6914652461, 6960900641

Offset: 1

Zak Seidov, May 26 2017

nonn

Lars Blomberg, Table of n, a(n) for n = 1..10000

Analogous sequences [with common difference in square brackets]: A033451 [6], A033447 [12], A033448 [18], A052242 [24], A052243 [30], A058252 [36], A058323 [42], A067388 [48], A259224 [54], A210683 [60].

More terms from Lars Blomberg, May 30 2017

A287550 Initial prime in set of 4 consecutive primes in arithmetic progression with difference 72.

Original entry on oeis.org

491525857, 1470227987, 2834347387, 4314407477, 4766711387, 6401372837, 6871241197, 8971400797, 10168905497, 11776429517, 11871902557, 14538547967, 14925896087, 15218517367, 15646776877, 15875854927, 17310026197, 17942416307, 18347931587, 19241492057, 19379888947

Offset: 1

Zak Seidov, May 26 2017

nonn

a(1)=491525857=A052239(12).

Chai Wah Wu, Table of n, a(n) for n = 1..100

Analogous sequences [with common difference in square brackets]: A033451 [6], A033447 [12], A033448 [18], A052242 [24], A052243 [30], A058252 [36], A058323 [42], A067388 [48], A259224 [54], A210683 [60]. Cf. A052239.

from gmpy2 import is_prime, next_prime
A287550_list, p = [], 2
q, r, s = p+72, p+144, p+216
while s <= 10**10:
    np = next_prime(p)
    if np == q and is_prime(r) and is_prime(s) and next_prime(q) == r and next_prime(r) == s:
        A287550_list.append(p)
    p, q, r, s = np, np+72, np+144, np+216 # Chai Wah Wu, Jun 03 2017

a(8)-a(21) from Chai Wah Wu, Jun 03 2017

A253216 Smallest of four primes in arithmetic progression with common difference 6 and digit sum prime.

Original entry on oeis.org

1091, 15791, 30091, 369991, 421691, 501191, 661091, 1101091, 1539991, 2042591, 2210291, 2542091, 2811191, 3351191, 3512291, 3864691, 4411391, 4675591, 5960791, 5992291, 5998691, 6884191, 6918391, 7516891, 8608591, 8697791, 9297091, 9622891, 9646291, 12013091

Offset: 1

K. D. Bajpai, Dec 29 2014

			a (1) = 1091: 1091 + 6 = 1097; 1097 + 6 = 1103; 1103 + 6 = 1109; all four are prime. Their digit sums 1+0+9+1 = 11; 1+0+9+7 = 17; 1+1+0+3 = 5 and 1+1+0+9 = 11 are also prime.
a(2) = 15791: 15791 + 6 = 15797; 15797 + 6 = 15803; 15803 + 6 = 15809; all four are prime. Their digit sums 1+5+7+9+1 = 23, 1+5+7+9+7 = 29, 1+5+8+0+3 = 17 and 1+5+8+0+9 = 23 are also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..2500

Cf. A000040, A033447, A062088, A253140.

A253216 = {}; Do[d = 6; k = Prime[n]; k1 = k + d; k2 = k + 2d; k3 = k + 3d; If[PrimeQ[k1] && PrimeQ[k2] && PrimeQ[k3] && PrimeQ[Plus @@ IntegerDigits[k]] && PrimeQ[Plus @@ IntegerDigits[k1]] && PrimeQ[Plus @@ IntegerDigits[k2]] && PrimeQ[Plus @@ IntegerDigits[k3]], AppendTo[A253216, k]], {n, 1000000}]; A253216
prQ[{a_,b_,c_,d_}]:=AllTrue[{b,c,d},PrimeQ]&&AllTrue[Total/@ (IntegerDigits/@ {a,b,c,d}),PrimeQ]; Select[#+{0,6,12,18}& /@Prime[Range[800000]],prQ][[All,1]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 26 2018 *)

for( n=1, 10^6, k=prime(n); k1=k+6; k2=k+12; k3=k+18; if(isprime(k1)&isprime(k2)&isprime(k3) &isprime(eval(Str(sumdigits(k)))) &isprime(eval(Str(sumdigits(k1)))) &isprime(eval(Str(sumdigits(k2)))) &isprime(eval(Str(sumdigits(k3)))), print1(k,", ")))

Definition corrected by Harvey P. Dale, May 26 2018

A253232 Smallest of five consecutive primes in arithmetic progression with common difference 90 and equal digit sums.

Original entry on oeis.org

61, 83, 89, 593, 1399, 2063, 2287, 2351, 2441, 3491, 5081, 5171, 5479, 6599, 9497, 12073, 16561, 17569, 21377, 23099, 23189, 28573, 29063, 32143, 36293, 36497, 36587, 39569, 49279, 61291, 62383, 65449, 66373, 71167, 72379, 75347, 81457, 88591, 92377, 94261, 104369

Offset: 1

K. D. Bajpai, Dec 29 2014

90 is the smallest common difference (d) to get a set of five consecutive primes in arithmetic progression {p, p+d, p+2d, p+3d, p+4d} having digit sums equal; for p < prime(10^5).

			a(1) = 61: 61+90 = 151; 151+90 = 241; 241+90 = 331; 331+90 = 421; all five are prime. Their digit sums 6+1 = 1+5+1 = 2+4+1 = 3+3+1 = 4+2+1 = 7 are all equal.
a(2) = 83: 83+90 = 173; 173+90 = 263; 263+90 = 353; 353+90 = 443; all five are prime. Their digit sums 8+3 = 1+7+3 = 2+6+3 = 3+5+3 = 4+4+3 = 11 are all equal.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000040, A033447, A062088, A253140.

A253232 = {}; Do[d = 90; k = Prime[n]; k1 = k + d; k2 = k + 2 d; k3 = k + 3 d; k4 = k + 4 d; s = Plus @@ IntegerDigits[k]; s1 = Plus @@ IntegerDigits[k1]; s2 = Plus @@ IntegerDigits[k2]; s3 = Plus @@ IntegerDigits[k3]; s4 = Plus @@ IntegerDigits[k4]; If[PrimeQ[k1] && PrimeQ[k2] && PrimeQ[k3] && PrimeQ[k4] && s == s1 && s1 == s2 && s2 == s3 && s3 == s4, AppendTo[A253232, k]], {n, 50000}]; A253232
cd90Q[p_]:=Module[{q=p+90,r=p+180,s=p+270,t=p+360},AllTrue[{p,q,r,s,t},PrimeQ] && Length[Union[Total/@(IntegerDigits/@{p,q,r,s,t})]]==1]; Select[ Prime[ Range[ 10000]],cd90Q] (* Harvey P. Dale, May 13 2022 *)

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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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A210683 Primes p such that p, p+60, p+120, p+180 are consecutive primes.

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A259224 Initial primes in sets of 4 consecutive primes with common gap 54.

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A058362 Initial primes of sets of 6 consecutive primes in arithmetic progression.

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A253140 Smallest of three consecutive primes in arithmetic progression with common difference 24 and digit sum prime.

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A248085 Initial prime of 4 primes in arithmetic progression with difference 12.

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A287547 Initial prime in set of 4 consecutive primes in arithmetic progression with difference 66.

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A287550 Initial prime in set of 4 consecutive primes in arithmetic progression with difference 72.

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