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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.
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
OFFSET
1,2
COMMENTS
First Differs from A354180 and A367802 at n = 113.
Also, exponentially 2^n-numbers that are squares.
Also, squares of exponentially 2^n-numbers.
FORMULA
a(n) = A138302(n)^2.
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=1} 1/p^(2^k)) = 1.62194750148969761827... .
MATHEMATICA
q[n_] := AllTrue[FactorInteger[n][[;; , 2]], # > 1 && # == 2^IntegerExponent[#, 2] &]; Select[Range[3300], # == 1 || q[#] &]
PROG
(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); }
CROSSREFS
Intersection of A001694 and A138302.
Intersection of A000290 and A138302.
Sequence in context: A179126 A354180 A367802 * A340674 A068879 A030152
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jan 26 2024
STATUS
approved