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The smaller of a pair of successive powerful numbers (A001694) without any prime number between them.
3

%I #34 Sep 16 2024 02:50:21

%S 8,25,32,121,288,675,1331,1369,1936,2187,2700,3125,5324,6724,9800,

%T 10800,12167,15125,32761,39200,48668,70225,79507,88200,97336,107648,

%U 143641,156800,212521,228484,235224,280900,312481,332928,456968,465124,574564,674028,744769,829921,830297,857476,877952,940896

%N The smaller of a pair of successive powerful numbers (A001694) without any prime number between them.

%H Amiram Eldar, <a href="/A240591/b240591.txt">Table of n, a(n) for n = 1..373</a> (terms below 10^15; terms 1..103 from Amiram Eldar, terms 104..235 from Chai Wah Wu)

%e 25 is in the sequence because A001694(6)=25, A001694(7)=27, without primes between them.

%t Select[Partition[Join[{1},Select[Range[10^6],Min@FactorInteger[#][[All, 2]]> 1&]],2,1],PrimePi[#[[1]]]==PrimePi[#[[2]]]&][[All,1]] (* _Harvey P. Dale_, Mar 28 2018 *)

%o (PARI)

%o ispowerful(n)={local(h);if(n==1,h=1,h=(vecmin(factor(n)[, 2])>1));return(h)}

%o nextpowerful(n)={local(k);k=n+1;while(!ispowerful(k),k+=1);return(k)}

%o {for(i=1,10^6,if(ispowerful(i),if(nextprime(i)>=nextpowerful(i),print1(i, ", "))))}

%o (Python)

%o from itertools import count, islice

%o from math import isqrt

%o from sympy import mobius, integer_nthroot, nextprime

%o def A240591_gen(): # generator of terms

%o def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))

%o def bisection(f,kmin=0,kmax=1):

%o while f(kmax) > kmax: kmax <<= 1

%o while kmax-kmin > 1:

%o kmid = kmax+kmin>>1

%o if f(kmid) <= kmid:

%o kmax = kmid

%o else:

%o kmin = kmid

%o return kmax

%o def f(x):

%o c, l, j = x-squarefreepi(integer_nthroot(x,3)[0]), 0, isqrt(x)

%o while j>1:

%o k2 = integer_nthroot(x//j**2,3)[0]+1

%o w = squarefreepi(k2-1)

%o c -= j*(w-l)

%o l, j = w, isqrt(x//k2**3)

%o return c+l

%o m = 1

%o for n in count(2):

%o k = bisection(lambda x:f(x)+n,m,m)

%o if nextprime(m) > k:

%o yield m

%o m = k

%o A240591_list = list(islice(A240591_gen(),30)) # _Chai Wah Wu_, Sep 14 2024

%Y Supersequence of A060355.

%Y Cf. A001694, A240590.

%K nonn

%O 1,1

%A _Antonio Roldán_, Apr 08 2014