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A257928 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. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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).

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(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).

MATHEMATICA

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}]

PROG

(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 */

CROSSREFS

Cf. A000040, A000720, A257364, A257938.

Sequence in context: A222464 A095389 A217518 * A206611 A298257 A298928

Adjacent sequences:  A257925 A257926 A257927 * A257929 A257930 A257931

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jul 13 2015

STATUS

approved

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Last modified May 16 10:48 EDT 2021. Contains 343941 sequences. (Running on oeis4.)