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A260078 Least positive integer k such that prime(k*n)-1+(prime(h*n)-1) = prime(i*n)-1 and prime(k*n)-1-(prime(h*n)-1) = prime(j*n)-1 for some positive integers h,i,j. 4
3, 3, 15, 5, 25, 29, 32, 20, 41, 87, 17, 61, 18, 100, 58, 10, 82, 82, 45, 74, 166, 20, 28, 338, 18, 35, 159, 290, 64, 29, 353, 311, 75, 41, 42, 492, 107, 155, 77, 364, 100, 330, 145, 474, 502, 332, 227, 553, 238, 92, 121, 597, 338, 339, 452, 164, 239, 832, 221, 243 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Conjecture: a(n) exists for any n > 0. In general, if m and n > 0 are integers with gcd(6,m) = 1, then the set {prime(k*n)+m: k = 1,2,3,...} contains two distinct elements x and y with x+y and x-y also in the set.

REFERENCES

Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..200

Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

EXAMPLE

a(2) = 3 since prime(3*2)-1+(prime(2*2)-1) = 12+6 = 18 = prime(4*2)-1, and prime(3*2)-1-(prime(2*2)-1) = 12-6 = 6 = prime(2*2)-1.

a(3) = 15 since prime(15*3)-1+(prime(12*3)-1) = 196+150 = 346 = prime(23*3)-1, and prime(15*3)-1-(prime(12*3)-1) = 196 -150 = 46 = prime(5*3)-1.

a(200) = 3319 since prime(3319*200)-1+(prime(2821*200)-1) = 9987120+8389110 = 18376230 = prime(5869*200)-1, and prime(3319*200)-1-(prime(2821*200)-1) = 9987120-8389110 = 1598010 = prime(605*200)-1.

MATHEMATICA

f[n_]:=Prime[n]-1

PQ[n_, p_]:=PrimeQ[p]&&Mod[PrimePi[p], n]==0

Do[k=0; Label[bb]; k=k+1; Do[If[PQ[n, f[k*n]+f[j*n]+1]&&PQ[n, f[k*n]-f[j*n]+1], Goto[aa]], {j, 1, k-1}]; Goto[bb];

Label[aa]; Print[n, " ", k]; Continue, {n, 1, 60}]

CROSSREFS

Cf. A000040, A257926, A257938.

Sequence in context: A100347 A165405 A179857 * A163590 A114320 A185138

Adjacent sequences:  A260075 A260076 A260077 * A260079 A260080 A260081

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jul 15 2015

STATUS

approved

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Last modified June 15 20:50 EDT 2019. Contains 324145 sequences. (Running on oeis4.)