A367804 Numbers that are both exponentially odd (A268335) and exponentially evil (A262675).

1, 8, 27, 32, 125, 216, 243, 343, 512, 864, 1000, 1331, 1944, 2197, 2744, 3125, 3375, 4000, 4913, 6859, 7776, 9261, 10648, 10976, 12167, 13824, 16807, 17576, 19683, 24389, 25000, 27000, 29791, 30375, 32768, 35937, 39304, 42592, 42875, 50653, 54872, 59319, 64000

Offset: 1

Author

Amiram Eldar, Dec 01 2023

Keywords

• nonn
• easy
• base

Comments

Numbers whose prime factorization contains only exponents that are odd evil numbers (A129771).

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 A262675 and A268335.

Cf. A367801, A367802, A367803.

Cf. A129771.

Programs

Mathematica

q[n_] := OddQ[n] && EvenQ[DigitCount[n, 2, 1]]; Select[Range[150], #== 1 || AllTrue[FactorInteger[#][[;;, 2]], q] &]

PARI

is(n) = {my(f = factor(n)); for (i = 1, #f~, if(!(f[i, 2]%2) || hammingweight(f[i, 2])%2, return (0))); 1;}

Formula

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=1} 1/p^A129771(k)) = Product_{p prime} f(1/p) = 1.22183814098622400889..., where f(x) = 1 + (2*x/(1-x^2) + Product_{k>=0} (1 - x^(2^k)) - Product_{k>=0} (1 - (-x)^(2^k)))/4.