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Prime powers such that the 4th power of the number of divisors is not smaller than the number itself.
5

%I #7 Jul 15 2017 11:05:24

%S 2,3,4,5,7,8,9,11,13,16,25,27,32,49,64,81,125,128,243,256,512,625,729,

%T 1024,2048,2187,4096,6561,8192,16384,32768,65536

%N Prime powers such that the 4th power of the number of divisors is not smaller than the number itself.

%C For any integers n, d[n]^4>n should form finite albeit very large sequence.

%F p^w<=(w+1)^4 i.e. p<=(w+1)^(4/w) restricts possible primes and their exponents

%e Equality holds in 12 cases: n=6561=3^8,d[n]=9 and d^4=9^4=3^8=n n=625,d[n]=5, so d^4=n

%t Select[Select[Range[2^16], PrimePowerQ], DivisorSigma[0, #]^4 >= # &] (* _Michael De Vlieger_, Jul 15 2017 *)

%Y A000005, A034884, A035033-A035035.

%K fini,full,nonn

%O 1,1

%A _Labos Elemer_, Aug 18 2000