A367801 Numbers that are both exponentially odd (A268335) and exponentially odious (A270428).

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106

Offset: 1

Author

Amiram Eldar, Dec 01 2023

Keywords

• nonn
• easy
• base

Comments

First differs from its subsequence A005117 at n = 79: a(79) = 128 is not a squarefree number.

First differs from A077377 at n = 63, and from A348506 at n = 68.

Numbers whose prime factorization contains only exponents that are odd odious numbers (A092246).

The asymptotic density of this sequence is Product_{p prime} f(1/p) = 0.61156148494581943994..., where f(x) = (1-x) * (1 + x/(2*(1-x^2)) + (Product_{k>=0} (1-(-x)^(2^k)) - Product_{k>=0} (1-x^(2^k))))/2.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..12232 (terms below 20000)
• Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175 (2016), pp. 385-395.

Crossrefs

Intersection of A268335 and A270428.

Cf. A367802, A367803, A367804.

Subsequences: A005117, A092759.

Cf. A092246.

Cf. A077377, A348506.

Programs

Mathematica

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

PARI

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