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A035095
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Smallest prime of form k*p(n) + 1, the arithmetic progressions of prime differences.
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9
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3, 7, 11, 29, 23, 53, 103, 191, 47, 59, 311, 149, 83, 173, 283, 107, 709, 367, 269, 569, 293, 317, 167, 179, 389, 607, 619, 643, 1091, 227, 509, 263, 823, 557, 1193, 907, 1571, 653, 2339, 347, 359, 1087, 383, 773, 3547, 797, 2111, 2677, 5449, 2749, 467
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Note that both the terms of this sequence and increments in the arithmetic progressions are primes.
This is one possible generalization of "the least prime problem in special arithmetic progressions" when n in the nk+1 form is replaced by n-th prime number.
Smallest numbers m such that largest prime-factor of Phi[m]=p(n), the n-th prime seems to be also prime number and identical to n-th of A035095. See A068211, A068212, A065966: Min[x : A068211(x)=p(n)]=A035095(n); e.g. Phi[a(7)]=Phi[103]=2.3.17 of which 17=p(7) is the largest prime-factor, arising first here.
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REFERENCES
| E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Bd 1 (reprinted Chelsea 1953)
E. C. Titmarsh, A divisor problem Renc Circ Math Palermo v 54 (1930) pp 414-429
P. Turan, Uber Primzahlen der arithmetischen Progression, Acta Sci Math. (Szeged) v8 (1936/37) pp 226-235
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LINKS
| T. D. Noe, Table of n, a(n) for n = 1..10000
P. Erdos, On some application of Brun's method, Acta Sci. Math (Szeged), v. 13, 1949 pp 57-63
A. Granville and C. Pomerance, On the least prime in certain arithmetic progressions
A. Granville and C. Pomerance, On the least prime in certain arithmetic progressions J. Lond Math Soc s2-41 (2) (1990) 193-200
D. R. Heath-Brown, almost-primes in arithmetic progressions in short intervals, Math Proc Cambr. Phil Soc v 83 (1978) pp 357-375
D. R. Heath-Brown, Siegel zeros and the least prime in arithmetic progression, Quart. J. of Math 41 (49) (1990) 405-418
H.-J. Kanold, Uber Primzahlen in arithmetischen Folgen, Math. Ann. v 156 (1964) pp 393-395
U. V. Linnik, On the least prime in an arithmetic progression. I. The basic theorem, Rec. Math (N.S.) v 15 (57) (1944), pp 139-178. MR0012111
C. Pomerance, A note on the least prime in an arithmetic progression J. Number Theory 12 (2) (1980) 218-223
K. Prachar, Uber die kleinste Primzahl in einer arithmetischen Reihe, J Reine Angew Math. 206 (1961) p 3-4
A. Schinzel, Remark on the paper of K. Prachar Uber die kleinste.., J. Reine Angew Math. v 210 (1962) pp 122-122
S. S. Wagstaff, Jr., The irregular primes to 125000, Math. Comp. v 32 (1978) pp 583-591.
S. S. Wagstaff, Jr, Greatest of the Least Primes in Arithmetic Progressions Having a Given Modulus, Math. Comp. 33 (147) (1979) pp 1073-1080
Index entries for sequences related to primes in arithmetic progressions
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FORMULA
| According to a long-standing conjecture (see the Wagstaff reference), a(n) <= prime(n)^2+1. This is sufficient to imply that a(n) is the smallest prime such that greatest prime divisor of a(n)-1 is prime(n), the n-th prime: A006530(a(n)-1)=A000040(n). This in turn is sufficient to imply that no value occurs twice in this sequence.
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EXAMPLE
| a(8) = 191 because in the prime(8)k+1 = 19k+1 sequence 191 is the smallest prime.
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MATHEMATICA
| a[n_] := (p = Prime[n]; r = 1; While[! PrimeQ[r], r += p]; r); Table[a[n], {n, 1, 51}] (* From Jean-François Alcover, Sep 20 2011, after PARI *)
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PROG
| (PARI) a(n)=local(p, r); p=prime(n); r=1; while(!isprime(r), r+=p); r
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CROSSREFS
| Cf. A034694, A032448, A006530, A006093, A035096, A000040, A019434, A058383.
Cf. A068211, A068212, A065966, A000010, A070844-A070858, A061092.
Sequence in context: A051202 * A066674 A125878 A126112 A194373 A156210
Adjacent sequences: A035092 A035093 A035094 * A035096 A035097 A035098
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KEYWORD
| nonn
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AUTHOR
| Labos E. (labos(AT)ana.sote.hu)
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EXTENSIONS
| Edited by Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Jun 18 2010
Minor edits by N. J. A. Sloane, Jun 27 2010
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