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A260197
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Least prime p such that pi(p*n) = prime(q*n) for some prime q, where pi(x) denotes the number of primes not exceeding x.
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3
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5, 277, 29, 17, 43, 103, 53, 31, 1571, 3089, 37, 593, 881, 3023, 277, 9257, 47, 1949, 9137, 311, 17011, 1039, 53, 59, 2153, 15331, 3617, 631, 44867, 61, 17351, 661, 821, 2339, 683, 1201, 34759, 62687, 20327, 59369, 71, 883, 40189, 9187, 1879, 7669, 2767, 3931, 8867, 8081, 79, 12401, 139, 4787, 6367, 277, 2903, 23671, 32839, 3659
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OFFSET
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1,1
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COMMENTS
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Conjecture: a(n) exists for any n > 0. Also, for any n > 0, there are primes p and q such that pi(p*n) = q*n.
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REFERENCES
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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.
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LINKS
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EXAMPLE
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a(1) = 5 since pi(5*1) = 3 = prime(2*1) with 2 and 5 both prime.
a(2) = 277 since pi(277*2) = 101 = prime(13*2) with 13 and 277 both prime.
a(10) = 3089 since pi(3089*10) = 3331 = prime(47*10) with 47 and 3089 both prime.
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MATHEMATICA
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PQ[n_, p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]/n]
Do[k=0; Label[aa]; k=k+1; If[PQ[n, PrimePi[Prime[k]*n]], Goto[bb], Goto[aa]]; Label[bb]; Print[n, " ", Prime[k]]; Continue, {n, 1, 60}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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