Amiram Eldar, <a href="/A369567/b369567_1.txt">Table of n, a(n) for n = 1..10000</a>

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

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

Amiram Eldar, <a href="/A369567/b369567_1.txt">Table of n, a(n) for n = 1..10000</a>

allocated for Amiram EldarPowerful exponentially 2^n-numbers: numbers whose prime factorization contains only exponents that are powers of 2 that are larger than 1.

1, 4, 9, 16, 25, 36, 49, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 625, 676, 784, 841, 900, 961, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2704, 2809, 3025, 3249

1,2

First Differs from A354180 and A367802 at n = 113.

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

Also, squares of exponentially 2^n-numbers.

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

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

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

(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); }

Intersection of A001694 and A138302.

Intersection of A000290 and A138302.

Cf. A354180, A367802.

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Amiram Eldar, Jan 26 2024

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