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A257928
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Least prime p such that pi(p*n) = pi(q*n)*pi(r*n) for some primes q and r with p > q > r, where pi(x) denotes the number of primes not exceeding x.
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7
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13, 7, 13, 67, 19, 79, 47, 193, 107, 41, 229, 179, 383, 281, 173, 1327, 193, 701, 1429, 211, 113, 73, 1093, 83, 1447, 659, 197, 719, 331, 761, 1171, 2269, 467, 509, 863, 113, 643, 577, 563, 379, 607, 1291, 283, 3593, 2549, 881, 1523, 4663, 2657, 3583, 8807, 683, 2251, 863, 8929, 163, 6737, 2459, 4919, 6553
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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 each positive integer n there are distinct primes p, q and r such that pi(p*n) = pi(q*n) + pi(r*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) = 13 since 3, 5 and 13 are distinct primes with pi(13*1) = 6 = 2*3 = pi(3*1)*pi(5*1).
a(200) = 105227 since 19, 113 and 105227 are distinct primes with pi(105227*200) = 1332672 = 528*2524 = pi(19*200)*pi(113*200).
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MATHEMATICA
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f[n_]:=PrimePi[n]
Do[k=0; Label[bb]; k=k+1; Do[Do[If[f[Prime[k]*n]==f[Prime[i]*n]*f[Prime[j]*n], Goto[aa]]; If[f[Prime[k]*n]<f[Prime[i]*n]*f[Prime[j]*n], Goto[cc]]; Continue, {i, 1, j-1}]; Label[cc]; Continue, {j, 1, k-1}]; Goto[bb];
Label[aa]; Print[n, " ", Prime[k]]; Continue, {n, 1, 60}]
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PROG
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(PARI) a(n)={my(i, j, k=3); while(1, for(j=2, k-1, for(i=1, j-1, if(primepi(prime(k)*n) == primepi(prime(i)*n)*primepi(prime(j)*n), break(3)); )); k++); return(prime(k)); } main(size)={return(vector(size, n, a(n))); } /* Anders Hellström, Jul 13 2015 */
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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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