A006560
Smallest starting prime for n consecutive primes in arithmetic progression.
Original entry on oeis.org
2, 2, 3, 251, 9843019, 121174811
Offset: 1
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);
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Jens Kruse Andersen, The smallest known CPAP-k.
- Thomas Bloom, Let k >= 3. Are there k consecutive primes in arithmetic progression?, Erdős Problems.
- Chris K. Caldwell, Consecutive Primes in Arithmetic Progression
- Harvey Dubner and Harry Nelson, Seven consecutive primes in arithmetic progression, Math. Comp., 66 (1997) 1743-1749. MR 98a:11122.
- H. Dubner, T. Forbes, N. Lygeros, M. Mizony, H. Nelson, and P. Zimmermann, Ten consecutive primes in arithmetic progression, Math. Comp., Vol. 71, No. 239 (2002) 1323-1328.
- Daniel Forgues, Wiki about consecutive primes in arithmetic progression.
- L. J. Lander and T. R. Parkin, Consecutive primes in arithmetic progression, Math. Comp., Vol. 21, No. 99 (1967) p 489.
- Terence Tao, Erdős problem database, see no. 141.
- Manfred Toplic, The nine and ten primes project, 2004.
- Index entries for sequences related to primes in arithmetic progressions
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.
-
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 *)
A059044
Initial primes of sets of 5 consecutive primes in arithmetic progression.
Original entry on oeis.org
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
Offset: 1
Harvey Dubner (harvey(AT)dubner.com), Dec 18 2000
- David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Rev. ed. 1997), p. 181.
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.
-
Select[Partition[Prime[Range[14000000]],5,1],Length[Union[ Differences[ #]]]==1&] (* Harvey P. Dale, Jun 22 2013 *)
-
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
Edited (definition clarified, cross-references corrected and extended) by
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
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
Comment split off from Name (to clarify definition) by
M. F. Hasler, Oct 27 2018
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