|
| |
|
|
A260140
|
|
Least prime p such that pi(p*n) = pi(q*n)^2 for some prime q, where pi(x) denotes the number of primes not exceeding x.
|
|
3
|
|
|
|
2, 5, 19, 3187, 11, 2251, 12149, 19, 239, 23761, 61, 157, 8419, 10973, 1117, 9601, 58741, 37, 53359, 14533, 1063, 934811, 78487, 27647, 1249, 720221, 1616077, 30091, 5501, 131627, 2003, 67, 677, 1313843, 45413, 273943, 127241, 19661, 188317, 811, 33863, 17789, 109073, 602269, 125201, 6424897, 441647, 2512897, 2909, 836471
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,1
|
|
|
COMMENTS
|
Conjecture: a(n) exists for any n > 0. In general, for any integers a,b,c,n with a > 0 and n > 0, there are two elements x and y of the set {pi(p*n): p is prime} with a*x^2+b*x+c = y.
A supplement to the conjecture: For any integers b,c,n with b > 0 and n > 0, we have b*x+c = y for some elements x and y of the set {pi(p*n): p is prime}. - Zhi-Wei Sun, Aug 02 2015
|
|
|
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
|
|
|
|
EXAMPLE
|
a(1) = 2 since pi(2*1) = 1^2 = pi(2*1)^2 with 2 prime.
a(4) = 3187 since pi(3187*4) = 1521 = 39^2 = pi(43*4)^2 with 43 and 3187 both prime.
a(72) = 25135867 since pi(25135867*72) = 89321401 = 9451^2 = pi(1367*72)^2 with 1367 and 25135867 both prime.
a(84) = 106788581 since pi(106788581*84) = 410224516 = 20254^2 = prime(2713*84)^2 with 2713 and 106788581 both prime.
|
|
|
MATHEMATICA
|
SQ[n_]:=IntegerQ[Sqrt[n]]
f[n_]:=PrimePi[n]
Do[k=0; Label[bb]; k=k+1; If[SQ[f[Prime[k]*n]]==False, Goto[bb]]; Do[If[Sqrt[f[Prime[k]*n]]==f[Prime[j]*n], Goto[aa]]; If[Sqrt[f[Prime[k]*n]]<f[Prime[j]*n], Goto[bb]]; Continue, {j, 1, k}]; Goto[bb]; Label[aa]; Print[n, " ", Prime[k]]; Continue, {n, 1, 50}]
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
