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251, 111497, 74453, 1397609, 642427, 5321191, 23921299, 55410683
Smallest starting prime for n consecutive primes in arithmetic progression.
(Formerly M0927)
+10
41
2, 2, 3, 251, 9843019, 121174811
COMMENTS
The primes following a(5) and a(6) occur at a(n)+30*k, k=0..(n-1). a(6) was found by Lander and Parkin. The next term requires a spacing >= 210. The expected size is a(7) > 10^21 (see link). - Hugo Pfoertner, Jun 25 2004 It is conjectured that there are arithmetic progressions of n consecutive primes for any n.
Common differences of first and smallest AP of n >= 1 consecutive primes: {0, 1, 2, 6, 30, 30, >= 210, >= 210, >= 210, >= 210, >= 2310, ...} (End)
a(7) <= 71137654873189893604531, found by P. Zimmermann, cf. J. K. Andersen link. - Bert Dobbelaere, Jul 27 2022
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
EXAMPLE
First and smallest occurrence of n, n >= 1, consecutive primes in arithmetic progression:
a(1) = 2: (2) (degenerate arithmetic progression);
a(2) = 2: (2, 3) (degenerate arithmetic progression);
a(3) = 3: (3, 5, 7);
a(4) = 251: (251, 257, 263, 269);
a(5) = 9843019: (9843019, 9843049, 9843079, 9843109, 9843139);
a(6) = 121174811: (121174811, 121174841, 121174871, 121174901, 121174931, 121174961);
MATHEMATICA
Join[{2}, Table[SelectFirst[Partition[Prime[Range[691*10^4]], n, 1], Length[ Union[ Differences[ #]]] == 1&][[1]], {n, 2, 6}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 10 2019 *)
CROSSREFS
Cf. A054800: start of 4 consecutive primes in arithmetic progression (CPAP-4), A033451: start of CPAP-4 with common difference 6, A052239: start of first CPAP-4 with common difference 6n. Cf. A059044: start of 5 consecutive primes in arithmetic progression, A210727: CPAP-5 with common difference 60. Cf. A058362: start of 6 consecutive primes in arithmetic progression.
Initial primes of sets of 5 consecutive primes in arithmetic progression.
+10
17
9843019, 37772429, 53868649, 71427757, 78364549, 79080577, 98150021, 99591433, 104436889, 106457509, 111267419, 121174811, 121174841, 168236119, 199450099, 203908891, 207068803, 216618187, 230952859, 234058871, 235524781, 253412317, 263651161, 268843033, 294485363, 296239787
COMMENTS
Each set has a constant difference of 30, for all of the terms listed so far.
It is conjectured that there exist arbitrarily long sequences of consecutive primes in arithmetic progression. As of December 2000, the record is 10 primes.
The first CPAP-5 with common difference 60 starts at 6182296037 ~ 6e9, cf. A210727. This sequence consists of first members of pairs of consecutive primes in A054800 (see also formula): a(1..6) = A054800({1555, 4555, 6123, 7695, 8306, 8371}). Conversely, pairs of consecutive primes in this sequence yield a term of A058362, i.e., they start a sequence of 6 consecutive primes in arithmetic progression (CPAP-6): e.g., the nearby values a(12) = 121174811, a(13) = 121174841 = a(12) + 30 indicate such a term, whence A006560(6) = A058362(1) = a(12). The first CPAP-6 with common difference 60 starts at 293826343073 ~ 3e11, cf. A210727. Longer CPAP's must have common difference >= 210. - M. F. Hasler, Oct 26 2018 About 500 initial terms of this sequence are the same as for the sequence "First of 5 consecutive primes separated by gaps of 30". The first 10^4 terms of A052243 give 281 terms of this sequence (up to ~ 3.34e9) with the same formula as the one using A054800, but as the above comment says, this will miss terms beyond twice that range. - M. F. Hasler, Jan 02 2020
REFERENCES
David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Rev. ed. 1997), p. 181.
FORMULA
Found by exhaustive search for 5 primes in arithmetic progression with all other intermediate numbers being composite.
MATHEMATICA
Select[Partition[Prime[Range[14000000]], 5, 1], Length[Union[ Differences[ #]]]==1&] (* Harvey P. Dale, Jun 22 2013 *)
PROG
(PARI) A059044(n, p=2, c, g, P)={forprime(q=p+1, , if(p+g!=p+=g=q-p, next, q!=P+2*g, c=3, c++>4, print1(P-2*g, ", "); n--||break); P=q-g); P-2*g} \\ This does not impose the gap to be 30, but it happens to be the case for the first values. - M. F. Hasler, Oct 26 2018
CROSSREFS
Cf. A054800: start of 4 consecutive primes in arithmetic progression (CPAP-4). Cf. A033451, A033447, A033448, A052242, A052243, A058252, A058323, A067388: start of CPAP-4 with common difference 6, 12, 18, ..., 48. Cf. A052239: start of first CPAP-4 with common difference 6n. Cf. A058362: start of 6 consecutive primes in arithmetic progression. Cf. A006560: first prime to start a CPAP-n.
AUTHOR
Harvey Dubner (harvey(AT)dubner.com), Dec 18 2000
EXTENSIONS
Edited (definition clarified, cross-references corrected and extended) by M. F. Hasler, Oct 26 2018
Initial primes of sets of 6 consecutive primes in arithmetic progression.
+10
7
121174811, 1128318991, 2201579179, 2715239543, 2840465567, 3510848161, 3688067693, 3893783651, 5089850089, 5825680093, 6649068043, 6778294049, 7064865859, 7912975891, 8099786711, 9010802341, 9327115723, 9491161423, 9544001791, 10101930253, 10523406343, 13193702321
COMMENTS
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
PROG
(PARI) 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
CROSSREFS
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.
AUTHOR
Harvey Dubner (harvey(AT)dubner.com), Dec 18 2000
EXTENSIONS
Comment split off from Name (to clarify definition) by M. F. Hasler, Oct 27 2018
The smallest of four consecutive primes where all three gaps are perfect squares.
+10
4
255763, 604441, 651361, 884497, 913063, 1065133, 1320211, 1526191, 2130133, 2376721, 2907727, 2911933, 2974891, 3190597, 3603583, 3690151, 3707497, 3962941, 4209643, 4245643, 4706101, 5057671, 5155567, 5223187, 5260711, 5321191, 5325571, 5410627
COMMENTS
Gaps occur as (36,4,36), (4,36,36), etc., all with at least one of them equal to 36 thru primes of 10^9.
A gap of 16 is first involved in 2376721 and 4706101, a gap of 64 first in 4245643, 5710531 and 21953641.
a(26) = 5321191 = A052239(6) = A058252(1) is the first term to be followed by three equal gaps, i.e., to start a sequence of consecutive primes in arithmetic progression (CPAP-4). - M. F. Hasler, Nov 06 2018
EXAMPLE
a(2) = 604441, the smallest of the consecutive primes 604441, 604477, 604481, 604517, with gaps of 36, 4 and 36, all perfect squares.
MATHEMATICA
PerfectSquareQ[n_] := JacobiSymbol[n, 13] =!= -1 && JacobiSymbol[n, 19] =!= -1 && JacobiSymbol[n, 17] =!= -1 && JacobiSymbol[n, 23] =!= -1 && IntegerQ[Sqrt[n]]; t = {}; n = 3; p1 = 1; p2 = 2; p3 = 3; p4 = 5; While[Length[t] < 30, n++; p1 = p2; p2 = p3; p3 = p4; p4 = Prime[n]; If[PerfectSquareQ[p2 - p1] && PerfectSquareQ[p3 - p2] && PerfectSquareQ[p4 - p3], AppendTo[t, p1]]]; t (* T. D. Noe, Jul 09 2013 *) Transpose[Select[Partition[Prime[Range[400000]], 4, 1], And@@IntegerQ/@ Sqrt[ Differences[#]]&]][[1]] (* Harvey P. Dale, Mar 24 2014 *)
Initial prime in set of 4 consecutive primes in arithmetic progression with difference 90.
+10
2
4480114337, 11598050027, 16107995833, 19544227597, 19921527331, 26594502101, 29300624819, 29558893643, 30312236239, 30570499049, 31578774413, 38072416381, 38554165297, 43191763313, 44924731961, 45264825169, 45870718093, 46642295699, 47647807219, 48621561623
Initial prime in set of 4 consecutive primes in arithmetic progression with difference 78.
+10
2
998051413, 4976990483, 5038640083, 5280622483, 5901887533, 6840045103, 8733200093, 11838052453, 11978292923, 12348742883, 12844988753, 13009792643, 13658681843, 13849573433
MATHEMATICA
Select[Partition[Prime[Range[7*10^8]], 4, 1], Differences[#]=={78, 78, 78}&][[All, 1]] (* Harvey P. Dale, Mar 17 2018 *)
PROG
(PARI) p=2; q=3; r=5; forprime(s=7, , if(s-r==78 && r-q==78 && q-p==78, print1(p", ")); p=q; q=r; r=s) \\ Charles R Greathouse IV, May 27 2017
Initial prime in set of 4 consecutive primes in arithmetic progression with difference 72.
+10
2
491525857, 1470227987, 2834347387, 4314407477, 4766711387, 6401372837, 6871241197, 8971400797, 10168905497, 11776429517, 11871902557, 14538547967, 14925896087, 15218517367, 15646776877, 15875854927, 17310026197, 17942416307, 18347931587, 19241492057, 19379888947
PROG
(Python)
from gmpy2 import is_prime, next_prime
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:
p, q, r, s = np, np+72, np+144, np+216 # Chai Wah Wu, Jun 03 2017
Initial prime in set of 4 consecutive primes in arithmetic progression with difference 84.
+10
1
2060959049, 5179429429, 5817066779, 12956411939, 13177258649, 13180695739, 13421782139, 14203265309, 14388905039, 14550841619, 14837398039, 16968792109, 20244234709
Smallest primes followed by n consecutive (unequal) prime differences, each divided by 6.
+10
0
23, 47, 251, 1889, 1741, 19471, 118801, 148537, 148531, 406951, 1820111, 2339041, 40727657, 19725473, 73451737, 232301497, 400414121, 1057859471, 489144599, 1444257673, 766319189
COMMENTS
This is a "modular arithmetic progression" of successive primes, modulo 6.
EXAMPLE
n=1 and 23 is followed by d=6 to give 29, a prime; n=5 a(5)=1741 is followed by {d}={6,6,6,18,6} and results in {1741,1747,1753,1759,1777,1783} consecutive prime sequence, while a(10)=406951 prime is followed by {18,12,12,30,24,12,24,36,18,12} consecutive d-pattern.
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