A107655 a(n) is the smallest number m greater than 1 such that phi(m) = d(m)^n, where d(m) is number of positive divisors of m; if there is no such m, a(n)=1.

3, 5, 85, 17, 1285, 4369, 559876, 257, 327685, 1114129, 1114521441417, 16843009, 160490068541289, 1925878801139721, 23110536763219977, 65537, 3327917287071744009, 39934999967815157769, 479219999336720898057, 5750639996603165650953, 69007679885506346588169, 828092158571811231498249, 9937105900443065378930697

Offset: 1

Author

Farideh Firoozbakht, Jun 06 2005

Keywords

• nonn

Comments

For n=0,1,2,3, and 4, a(2^n) = A000215(n), the n-th Fermat prime.

Conjecture: A000005(a(n)) <= 12 for all n. [Max Alekseyev, May 07 2010]

This conjecture holds throughout the first 102 terms. - David A. Corneth, Jun 14 2020

Examples

			a(10) = 1114129 because phi(1114129) = d(1114129)^10 and 1114129 is the smallest number m greater than 1 that phi(m) = 1048576 = 4^10 = d(m)^10.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..100
• Max Alekseyev, PARI scripts for various problems (see invphitau there).

Crossrefs

Cf. A000005, A000215, A033844.

Programs

PARI

a(n)=res = oo; for(i=2, oo, if(i^n > res, return(res)); c=invphitau(i^n,i); if(#c>0, res=c[1])) \\ for invphitau, see Alekseyev link \\ David A. Corneth, Jun 14 2020

Extensions

Terms a(11) onward from Max Alekseyev, May 07 2010

Terms a(20)-a(23), offset corrected by David A. Corneth, Jun 14 2020