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Least positive integer m with pi(m*n) = m + n, where pi(x) denotes the number of primes not exceeding x.
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%I #41 Jun 06 2024 12:32:49

%S 9,7,6,998,5,5,5,5,5,5,636787,1617099,4124188,10553076,5,5,179992154,

%T 465769460,1208198239,3140421185,5,5,5,145935688930,5,5,5,5,5,5,5,5,5

%N Least positive integer m with pi(m*n) = m + n, where pi(x) denotes the number of primes not exceeding x.

%C The author proved that a(n) exists for every n >= 5.

%C a(39) = a(41) = 5. - _Chai Wah Wu_, Jun 06 2024

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1409.5685">A new theorem on the prime-counting function</a>, arXiv:1409.5685 [math.NT], 2014-2017.

%e a(5) = 9 since pi(5*9) = 14 = 5 + 9, and pi(5*m) = 5 + m for no m < 9.

%t Do[m=1;Label[aa];If[PrimePi[n*m]==m+n,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,5,21}]

%t Table[m = 1; While[PrimePi[m*n] != m + n, m++]; m, {n, 5, 14}] (* _Robert Price_, Mar 20 2019 *)

%Y Cf. A247601, A247602, A247603, A247604, A000720, A281197.

%K nonn,more

%O 5,1

%A _Zhi-Wei Sun_, Sep 21 2014

%E a(22)-a(37) from _Chai Wah Wu_, May 03 2018