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A270337 Composite numbers equal to the number of divisors of one of their powers. 2


%S 9,25,28,40,45,49,81,121,153,169,225,289,325,343,361,441,529,625,640,

%T 841,961,976,1089,1225,1369,1521,1681,1849,2133,2197,2209,2401,2541,

%U 2601,2809,3025,3249,3481,3721,4225,4489,4753,4761,4851,5041,5329,5929,6241,6348,6561,6859,6889

%N Composite numbers equal to the number of divisors of one of their powers.

%C Prime numbers are not considered since every prime p satisfies p = d(p^(p-1)), where d() represents the number of divisors.

%C In general, p^k = d((p^k)^((p^k-1)/k)) for any prime p and for any power k such that (p^k-1)/k is an integer.

%H Paolo P. Lava, <a href="/A270337/a270337.txt">First 50 terms with their powers</a>

%e 9 = d(9^4); 28 = d(28^3); 153 = d(153^8); etc.

%p with(numtheory): P:=proc(q) local a,k,n;

%p for n from 2 to q do if not isprime(n) then a:=tau(n); k:=0;

%p while a<n do k:=k+1; a:=tau(n^k); od; if n=a then print(n); fi; fi; od; end: P(10^6);

%t nn = 2000; Select[Select[Range@ nn, CompositeQ], Function[k, (SelectFirst[k^Range[nn/2], DivisorSigma[0, #] == k &] /. n_ /; MissingQ@ n -> 0) > 0]] (* _Michael De Vlieger_, Mar 17 2016, Version 10.2 *)

%Y Cf. A000005, A073049, A270389.

%K nonn,easy

%O 1,1

%A _Paolo P. Lava_, Mar 15 2016

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Last modified March 3 04:02 EST 2021. Contains 341756 sequences. (Running on oeis4.)