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A369567 Powerful exponentially 2^n-numbers: numbers whose prime factorization contains only exponents that are powers of 2 that are larger than 1.

%I A369567 #8 Jan 26 2024 08:35:07
%S A369567 1,4,9,16,25,36,49,81,100,121,144,169,196,225,256,289,324,361,400,441,
%T A369567 484,529,625,676,784,841,900,961,1089,1156,1225,1296,1369,1444,1521,
%U A369567 1681,1764,1849,1936,2025,2116,2209,2304,2401,2500,2601,2704,2809,3025,3249
%N A369567 Powerful exponentially 2^n-numbers: numbers whose prime factorization contains only exponents that are powers of 2 that are larger than 1.
%C A369567 First Differs from A354180 and A367802 at n = 113.
%C A369567 Also, exponentially 2^n-numbers that are squares.
%C A369567 Also, squares of exponentially 2^n-numbers.
%H A369567 Amiram Eldar, <a href="/A369567/b369567.txt">Table of n, a(n) for n = 1..10000</a>
%H A369567 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%H A369567 <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.
%F A369567 a(n) = A138302(n)^2.
%F A369567 Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=1} 1/p^(2^k)) = 1.62194750148969761827... .
%t A369567 q[n_] := AllTrue[FactorInteger[n][[;; , 2]], # > 1 && # == 2^IntegerExponent[#, 2] &]; Select[Range[3300], # == 1 || q[#] &]
%o A369567 (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 A369567 Intersection of A001694 and A138302.
%Y A369567 Intersection of A000290 and A138302.
%Y A369567 Cf. A354180, A367802.
%K A369567 nonn
%O A369567 1,2
%A A369567 _Amiram Eldar_, Jan 26 2024