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A367803
Exponentially evil squares.
4
1, 64, 729, 1024, 4096, 15625, 46656, 59049, 117649, 262144, 531441, 746496, 1000000, 1048576, 1771561, 2985984, 3779136, 4826809, 7529536, 9765625, 11390625, 16000000, 16777216, 24137569, 34012224, 47045881, 60466176, 64000000, 85766121, 113379904, 120472576, 148035889
OFFSET
1,2
COMMENTS
Numbers whose prime factorization contains only exponents that are even evil numbers (A125592).
Also, squares of exponentially evil numbers (A262675).
Also, numbers with an equal number of exponentially odious and exponentially evil divisors, i.e., numbers k such that A366901(k) = A366902(k). - Amiram Eldar, Feb 26 2024
LINKS
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175 (2016), pp. 385-395.
FORMULA
a(n) = A262675(n)^2.
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=1} 1/p^A125592(k)) = Product_{p prime} f(1/p) = 1.01833932269003592136..., where f(x) = (2/(1-x^2) + Product_{k>=0} (1 - x^(2^k)) + Product_{k>=0} (1 - (-x)^(2^k)))/4.
MATHEMATICA
evilQ[n_] := EvenQ[DigitCount[n, 2, 1]]; Select[Range[10^4]^2, #== 1 || AllTrue[FactorInteger[#][[;; , 2]], evilQ] &]
PROG
(PARI) isexpevil(n) = {my(f = factor(n)); for (i = 1, #f~, if(hammingweight(f[i, 2])%2, return (0))); 1; }
is(n) = issquare(n) && isexpevil(n);
CROSSREFS
Intersection of A000290 and A262675.
Sequence in context: A164345 A164337 A354178 * A161860 A195249 A223953
KEYWORD
nonn,easy,base
AUTHOR
Amiram Eldar, Dec 01 2023
STATUS
approved