A392741 - OEIS

%I #33 Mar 14 2026 09:43:33

%S 900,1764,4356,4900,6084,10404,11025,12100,12996,16900,19044,23716,

%T 27000,27225,28900,30276,33124,34596,36100,38025,44100,49284,52900,

%U 53361,56644,60516,65025,66564,70756,74088,74529,79524,81225,81796,84100,96100,101124,103684

%N Powers k^m, m > 1, squarefree k with more than 2 distinct prime factors.

%C Perfect powers of k in A350352.

%C This sequence is A303606 \ A303661.

%C Intersection of A000977 and A303606.

%C Superset of A162143, superset of A391320.

%H Michael De Vlieger, <a href="/A392741/b392741.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.

%F Sum_{n>=1} 1/a(n) = Sum_{k>=2} (zeta(k)/zeta(2*k) - P(k) - (P(k)^2 - P(2*k))/2 - 1) = 0.0038711835510255541315..., where P(k) is the prime zeta function. - _Amiram Eldar_, Mar 14 2026

%e Table of n, a(n) for select n:

%e     n        a(n)

%e   ----------------------------------------------

%e     1      900 =  30^2 = 2^2 * 3^2 *  5^2

%e     2     1764 =  42^2 = 2^2 * 3^2 *  7^2

%e     3     4356 =  66^2 = 2^2 * 3^2 * 11^2

%e     4     4900 =  70^2 = 2^2 * 5^2 *  7^2

%e     5     6084 =  78^2 = 2^2 * 3^2 * 13^2

%e     6    10404 = 102^2 = 2^2 * 3^2 * 17^2

%e     7    11025 = 105^2 = 3^2 * 5^2 *  7^2

%e    13    27000 =  30^3 = 2^3 * 3^3 *  5^3

%e    21    44100 = 210^2 = 2^2 * 3^2 *  5^2 * 7^2

%e   138   810000 =  30^4 = 2^4 * 3^4 *  5^4

%t nn = 100000; i = 1; MapIndexed[Set[S[First[#2]], #1] &, Select[Range@ Sqrt[nn], 2 < PrimeNu[#] == PrimeOmega[#] &]]; Union@ Reap[While[j = 2; While[S[i]^j < nn, Sow[S[i]^j]; j++]; j > 2, i++] ][[-1, 1]]

%o (Python)

%o from math import isqrt, prod

%o from sympy import primerange, integer_nthroot, primepi

%o from oeis_sequences.OEISsequences import bisection

%o def A392741(n):

%o     def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))

%o     def h(x): return sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,i)) for i in range(3,x.bit_length()))

%o     def f(x): return int(n+x-sum(h(integer_nthroot(x,i)[0]) for i in range(2,x.bit_length())))

%o     return bisection(f,n,n) # _Chai Wah Wu_, Mar 09 2026

%Y Cf. A000977, A001597, A001694, A072777, A120944, A126706, A131605, A162143, A286708, A303606, A303661, A350352, A375055, A390950, A391320.

%K nonn,easy

%O 1,1

%A _Michael De Vlieger_, Feb 28 2026