A247603 Least integer m > 0 with pi(m*n) = sigma(m), where sigma(m) is the sum of all positive divisors of m.

1, 2, 23, 61, 8, 22, 16, 12, 202, 386, 30, 36, 174, 10745, 1684, 2804, 1616, 40006, 6764, 996, 5775, 8131355, 19974, 11264, 4446, 27882, 4848, 32466, 162712, 532313373, 2341816, 30864, 14544, 63696, 2880, 390990, 135200, 133992, 1331840, 11621646, 117990

Offset: 2

Author

Zhi-Wei Sun, Sep 21 2014

Keywords

• nonn

Comments

Conjecture: a(n) exists for any n > 1.

Examples

			a(3) = 2 since pi(3*2) = 3 = sigma(2), and pi(3*1) = 2 > sigma(1) = 1.
		

Links

• Hiroaki Yamanouchi, Table of n, a(n) for n = 2..53
• Zhi-Wei Sun, A new theorem on the prime-counting function, arXiv:1409.5685, 2014.

Crossrefs

Cf. A000203, A000720, A247600, A247601, A247602, A247604, A247672, A247673.

Programs

Mathematica

Do[m=1;Label[aa];If[PrimePi[n*m]==DivisorSigma[1,m],Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];
Label[bb];Continue,{n,2,30}]

Extensions

a(31)-a(42) from Hiroaki Yamanouchi, Oct 04 2014