A369567 Powerful 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

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.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from exponents in factorization of n.

Index entries for sequences related to powerful 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.

Cf. A354180, A367802.

Sequence in context: A179126 A354180 A367802 * A340674 A068879 A030152

Adjacent sequences: A369564 A369565 A369566 * A369568 A369569 A369570

Keyword

nonn

Author

Amiram Eldar, Jan 26 2024

Status

approved