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A005115
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Let i, i+d, i+2d, ..., i+(n-1)d be an n-term arithmetic progression of primes; choose the one which minimizes the last term; then a(n) = last term i+(n-1)d.
(Formerly M0854)
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11
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2, 3, 7, 23, 29, 157, 907, 1669, 1879, 2089, 249037, 262897, 725663, 36850999, 173471351, 198793279, 4827507229, 17010526363, 83547839407, 572945039351, 6269243827111
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| In other words, smallest prime which is at the end of an arithmetic progression of n primes.
For the corresponding values of the first term and the common difference see A113827 and A093364. For the actual arithmetic progressions see A133277.
One may also minimize the common difference: this leads to A033189, A033188 and A113872.
One may also specify that the first term is the n-th prime and then minimize the common difference (or, equally, the last term): this leads to A088430 and A113834.
One may also ask for n consecutive primes in arithmetic progression: this gives A006560.
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REFERENCES
| H. Dubner and H. Nelson, Seven consecutive primes in arithmetic progression, Math. Comp., 66 (1997) 1743-1749. MR 98a:11122.
R. K. Guy, Unsolved Problems in Number Theory, A5.
A. Moran, P. Pritchard and A. Thyssen, Twenty-two primes in arithmetic progression, Math. Comp.64 (1995), no.211, 1337-1339.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| Jens Kruse Andersen, Primes in Arithmetic Progression Records [May have candidates for later terms in this sequence.]
Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions
Andrew Granville, Prime number patterns
Index entries for sequences related to primes in arithmetic progressions
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EXAMPLE
| n, AP, last term
1 2 2
2 2+j 3
3 3+2j 7
4 5+6j 23
5 5+6j 29
6 7+30j 157
7 7+150j 907
8 199+210j 1669
9 199+210j 1879
10 199+210j 2089
11 110437+13860j 249037
12 110437+13860j 262897
..........................
a(11)=249037 since 110437,124297,...,235177,249037 is an arithmetic progression of 11 primes ending with 249037 and it is the least number with this property.
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MATHEMATICA
| (* This program will generate the 4 to 12 terms to use a[n_] to generate term 13 or higher, it will have a prolonged run time. *) a[n_] := Module[{i, p, found, j, df, k}, i = 1; While[i++; p = Prime[i]; found = 0; j = 0; While[j++; df = 6*j; (p > ((n - 1)*df)) && (found == 0), found = 1; Do[If[! PrimeQ[p - k*df], found = 0], {k, 1, n - 1}]]; found == 0]; p]; Table[a[i], {i, 4, 12}]
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CROSSREFS
| For the associated gaps see A093364, for the initial terms see A113827. Cf. A006560, A096003.
Cf. A113830-A113834, A088430.
Sequence in context: A156615 A158054 A134412 * A113872 A120302 A093363
Adjacent sequences: A005112 A005113 A005114 * A005116 A005117 A005118
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KEYWORD
| nonn,nice,hard,more
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| a(11)-a(13) from Michael Somos, Mar 14 2004.
a(14) and corrected version of a(7) from Hugo Pfoertner (hugo(AT)pfoertner.org), Apr 27 2004
a(15)-a(17) from Don Reble (djr(AT)nk.ca), Apr 27 2004
a(18)-a(21) from Granville's paper, Jan 26, 2006
Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Jan 26 2006, Oct 17 2007
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