A175667 Smallest number m such that phi(m) = n*tau(m), with phi=A000010 and tau=A000005; a(n)=0 if no such m exists.

1, 5, 7, 34, 11, 13, 58, 17, 19, 55, 23, 65, 106, 29, 31, 85, 0, 37, 0, 41, 43, 115, 47, 119, 125, 53, 133, 145, 59, 61, 0, 388, 67, 274, 71, 73, 298, 0, 79, 187, 83, 203, 346, 89, 209, 235, 0, 97, 394, 101, 103, 169, 107, 109, 253, 113, 458, 295, 0, 287, 0, 0, 127, 514, 131

Offset: 1

Author

Enrique Pérez Herrero, Aug 05 2010

Keywords

• nonn

Comments

If p = 2*n+1 is a prime, and if n > 1 then a(n)=p.

From R. J. Mathar, Aug 07 2010: (Start)

First column in the array

1,3,8,10,18,24,30: A020488

5,9,15,28,40,72,84,90,120: A062516

7,21,26,56,70,78,108,126,168,210: A063469

34,45,52,102,140,156,252,360,420: A063470

11,33,88,110,198,264,330,

13,35,39,63,76,104,105,130,228,234,280,312,390,504,540,630,840,

58,98,174,294,

17,51,128,136,170,176,224,260,306,384,408,468,510,528,672,780,1260,

19,57,74,135,152,182,190,222,342,456,546,570,756,1080,

55,82,99,124,165,246,308,350,372,440,792,924,990,1050,1320,

23,69,184,230,414,552,690,

65,117,148,195,238,315,364,380,444,520,684,714,864,936,1092,1140,1170,1560,2520,

... (End)

Links

• Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A112954, A112955

Cf. A005384, A023212, A043297.

Programs

Mathematica

Table[SelectFirst[Range[10^5], EulerPhi@ # == n DivisorSigma[0, #] &] /.
k_ /; MissingQ@ k -> 0, {n, 120}] (* Michael De Vlieger, Aug 09 2017, Version 10.2 *)

Formula

From Enrique Pérez Herrero, Jan 01 2012: (Start)

If n > 1 then a(n) >= 2*n+1 or a(n)=0.

If p and q = 2*p+1 are both prime, A005384, then a(p) = 2*p+1.

If p > 3 and q = 4*p+1 are both prime, A023212, then a(p) = 8*p + 2 = 2*q.

If p > 2 is prime and both 2*p+1 and 4*p+1 are composite, A043297, then a(n)=0.

(End)

Extensions

More terms from R. J. Mathar, Aug 07 2010

Comment corrected by Enrique Pérez Herrero, Aug 12 2010