A367803 - OEIS

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

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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

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

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.

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

Cf. A001969, A125592.

Sequence in context: A164345 A164337 A354178 * A161860 A195249 A223953

Adjacent sequences: A367800 A367801 A367802 * A367804 A367805 A367806

KEYWORD

nonn,easy,base

AUTHOR

Amiram Eldar, Dec 01 2023

STATUS

approved