A281197 -id:A281197 - cp's OEIS frontend

A247600 Least positive integer m with pi(m*n) = m + n, where pi(x) denotes the number of primes not exceeding x.

9, 7, 6, 998, 5, 5, 5, 5, 5, 5, 636787, 1617099, 4124188, 10553076, 5, 5, 179992154, 465769460, 1208198239, 3140421185, 5, 5, 5, 145935688930, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 5

Zhi-Wei Sun, Sep 21 2014

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

a(39) = a(41) = 5. - Chai Wah Wu, Jun 06 2024

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

Zhi-Wei Sun, A new theorem on the prime-counting function, arXiv:1409.5685 [math.NT], 2014-2017.

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

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}]
Table[m = 1; While[PrimePi[m*n] != m + n, m++]; m, {n, 5, 14}] (* Robert Price, Mar 20 2019 *)

a(22)-a(37) from Chai Wah Wu, May 03 2018

A281196 Number of n such that A247600(n) = 5.

Original entry on oeis.org

9, 10, 11, 12, 13, 14, 19, 20, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 39, 41

Offset: 1

Seiichi Manyama, Jan 17 2017

Conjecture: This sequence has just 22 terms shown.

Sequence is finite and complete. Since pi(x) < 1.25506*x/log(x), it follows that for x > 106, pi(5*x) < x + 5. - Chai Wah Wu, May 01 2018, May 14 2020.

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

Cf. A000720, A247600, A281197.

pi(5*a(n)) = a(n) + 5.

A303910 Largest positive integer m with pi(m*n) = m + n, where pi(x) denotes the number of primes not exceeding x.

Original entry on oeis.org

41, 148, 409, 998, 2597, 6424, 15889, 39970, 100192, 251637, 637063, 1617110, 4124272, 10553278, 27066824, 69709557, 179992524, 465769748, 1208198402, 3140421406, 8179001739, 21338685037, 55762148901, 145935688930, 382465573384, 1003652346785, 2636913002779

Offset: 5

Chai Wah Wu, May 02 2018

nonn

a(n) >= A281197(n) >= A247600(n).

Cf. A247600, A281197.

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A247600 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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A281196 Number of n such that A247600(n) = 5.

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A303910 Largest positive integer m with pi(m*n) = m + n, where pi(x) denotes the number of primes not exceeding x.

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