A367803 Exponentially evil squares.

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

Author

Amiram Eldar, Dec 01 2023

Keywords

• nonn
• easy
• base

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

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175 (2016), pp. 385-395.

Crossrefs

Intersection of A000290 and A262675.

Cf. A366901, A366902, A367801, A367802, A367804.

Cf. A001969, A125592.

Programs

Mathematica

evilQ[n_] := EvenQ[DigitCount[n, 2, 1]]; Select[Range[10^4]^2, #== 1 || AllTrue[FactorInteger[#][[;;, 2]], evilQ] &]

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);

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.