A369567 - OEIS

%I #8 Jan 26 2024 08:35:07

%S 1,4,9,16,25,36,49,81,100,121,144,169,196,225,256,289,324,361,400,441,

%T 484,529,625,676,784,841,900,961,1089,1156,1225,1296,1369,1444,1521,

%U 1681,1764,1849,1936,2025,2116,2209,2304,2401,2500,2601,2704,2809,3025,3249

%N Powerful exponentially 2^n-numbers: numbers whose prime factorization contains only exponents that are powers of 2 that are larger than 1.

%C First Differs from A354180 and A367802 at n = 113.

%C Also, exponentially 2^n-numbers that are squares.

%C Also, squares of exponentially 2^n-numbers.

%H Amiram Eldar, <a href="/A369567/b369567.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.

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

%F a(n) = A138302(n)^2.

%F Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=1} 1/p^(2^k)) = 1.62194750148969761827... .

%t q[n_] := AllTrue[FactorInteger[n][[;; , 2]], # > 1 && # == 2^IntegerExponent[#, 2] &]; Select[Range[3300], # == 1 || q[#] &]

%o (PARI) is(n) = {my(e = factor(n)[, 2]); if(n == 1, 1, for(i = 1, #e, if(e[i] == 1 || e[i] >> valuation(e[i], 2) > 1, return(0))); 1);}

%Y Intersection of A001694 and A138302.

%Y Intersection of A000290 and A138302.

%Y Cf. A354180, A367802.

%K nonn

%O 1,2

%A _Amiram Eldar_, Jan 26 2024