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A325240 Numbers whose minimum prime exponent is 2.

%I A325240 #14 Oct 02 2024 12:25:35
%S A325240 4,9,25,36,49,72,100,108,121,144,169,196,200,225,288,289,324,361,392,
%T A325240 400,441,484,500,529,576,675,676,784,800,841,900,961,968,972,1089,
%U A325240 1125,1152,1156,1225,1323,1352,1369,1372,1444,1521,1568,1600,1681,1764,1800
%N A325240 Numbers whose minimum prime exponent is 2.
%C A325240 Or barely powerful numbers, a subset of powerful numbers A001694.
%C A325240 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose minimum multiplicity is 2 (counted by A244515).
%C A325240 Powerful numbers (A001694) that are not cubefull (A036966). - _Amiram Eldar_, Jan 30 2023
%H A325240 Amiram Eldar, <a href="/A325240/b325240.txt">Table of n, a(n) for n = 1..10000</a>
%F A325240 Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Product_{p prime} (1 + 1/(p^2*(p-1))) = A082695 - A065483 = 0.6038122832... . - _Amiram Eldar_, Jan 30 2023
%e A325240 The sequence of terms together with their prime indices begins:
%e A325240     4: {1,1}
%e A325240     9: {2,2}
%e A325240    25: {3,3}
%e A325240    36: {1,1,2,2}
%e A325240    49: {4,4}
%e A325240    72: {1,1,1,2,2}
%e A325240   100: {1,1,3,3}
%e A325240   108: {1,1,2,2,2}
%e A325240   121: {5,5}
%e A325240   144: {1,1,1,1,2,2}
%e A325240   169: {6,6}
%e A325240   196: {1,1,4,4}
%e A325240   200: {1,1,1,3,3}
%e A325240   225: {2,2,3,3}
%e A325240   288: {1,1,1,1,1,2,2}
%e A325240   289: {7,7}
%e A325240   324: {1,1,2,2,2,2}
%e A325240   361: {8,8}
%e A325240   392: {1,1,1,4,4}
%e A325240   400: {1,1,1,1,3,3}
%t A325240 Select[Range[1000],Min@@FactorInteger[#][[All,2]]==2&]
%o A325240 (PARI) is(n)={my(e=factor(n)[,2]); n>1 && vecmin(e) == 2; } \\ _Amiram Eldar_, Jan 30 2023
%o A325240 (Python)
%o A325240 from math import isqrt, gcd
%o A325240 from sympy import integer_nthroot, factorint, mobius
%o A325240 def A325240(n):
%o A325240     def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
%o A325240     def bisection(f,kmin=0,kmax=1):
%o A325240         while f(kmax) > kmax: kmax <<= 1
%o A325240         while kmax-kmin > 1:
%o A325240             kmid = kmax+kmin>>1
%o A325240             if f(kmid) <= kmid:
%o A325240                 kmax = kmid
%o A325240             else:
%o A325240                 kmin = kmid
%o A325240         return kmax
%o A325240     def f(x):
%o A325240         c, l = n+x, 0
%o A325240         j = isqrt(x)
%o A325240         while j>1:
%o A325240             k2 = integer_nthroot(x//j**2,3)[0]+1
%o A325240             w = squarefreepi(k2-1)
%o A325240             c -= j*(w-l)
%o A325240             l, j = w, isqrt(x//k2**3)
%o A325240         c -= squarefreepi(integer_nthroot(x,3)[0])-l
%o A325240         for w in range(1,integer_nthroot(x,5)[0]+1):
%o A325240             if all(d<=1 for d in factorint(w).values()):
%o A325240                 for y in range(1,integer_nthroot(z:=x//w**5,4)[0]+1):
%o A325240                     if gcd(w,y)==1 and all(d<=1 for d in factorint(y).values()):
%o A325240                         c += integer_nthroot(z//y**4,3)[0]
%o A325240         return c
%o A325240     return bisection(f,n,n**2) # _Chai Wah Wu_, Oct 02 2024
%Y A325240 Positions of 2's in A051904.
%Y A325240 Maximum instead of minimum gives A067259.
%Y A325240 Cf. A001221, A001222, A001358, A001694, A007774, A036966, A051903, A052485, A118914, A244515, A325241.
%Y A325240 Cf. A065483, A082695.
%K A325240 nonn
%O A325240 1,1
%A A325240 _Gus Wiseman_, Apr 15 2019