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Squares and cubes of primes.
12

%I #29 Aug 13 2024 11:21:15

%S 4,8,9,25,27,49,121,125,169,289,343,361,529,841,961,1331,1369,1681,

%T 1849,2197,2209,2809,3481,3721,4489,4913,5041,5329,6241,6859,6889,

%U 7921,9409,10201,10609,11449,11881,12167,12769,16129,17161,18769,19321,22201

%N Squares and cubes of primes.

%C Primitive elements for powerful numbers; every powerful is product of these numbers. The representation is not necessarily unique.

%F A178254(a(n)) = 2. - _Reinhard Zumkeller_, May 24 2010

%F Sum_{n>=1} 1/a(n) = P(2) + P(3) = 0.6270100593..., where P is the prime zeta function. - _Amiram Eldar_, Dec 21 2020

%t m=30000;Union[Prime[Range[PrimePi[m^(1/2)]]]^2,Prime[Range[PrimePi[m^(1/3)]]]^3] (* _Vladimir Joseph Stephan Orlovsky_, Apr 11 2011 *)

%t With[{nn=50},Take[Union[Flatten[Table[{n^2,n^3},{n,Prime[Range[ nn]]}]]],nn]] (* _Harvey P. Dale_, Feb 26 2015 *)

%o (PARI) for(n=1,40000,fm=factor(n);if(matsize(fm)[1]==1&(fm[1,2]==2||fm[1,2]==3),print1(n",")))

%o (PARI) is(n)=my(k=isprimepower(n)); k && k<4 \\ _Charles R Greathouse IV_, May 24 2013

%o (Python)

%o from math import isqrt

%o from sympy import primepi, integer_nthroot

%o def A168363(n):

%o def f(x): return n+x-primepi(isqrt(x))-primepi(integer_nthroot(x,3)[0])

%o m, k = n, f(n)

%o while m != k:

%o m, k = k, f(k)

%o return int(m) # _Chai Wah Wu_, Aug 09 2024

%Y Cf. A001694, A053810, A001248, A030078, A087797, A178254.

%K nonn

%O 1,1

%A _Franklin T. Adams-Watters_, Nov 23 2009