A367802 Exponentially odious squares.

1, 4, 9, 16, 25, 36, 49, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 625, 676, 784, 841, 900, 961, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2704, 2809, 3025, 3249

Offset: 1

Author

Amiram Eldar, Dec 01 2023

Keywords

• nonn
• easy
• base

Comments

First differs from A354180 at n = 226.

Numbers whose prime factorization contains only exponents that are even odious numbers (A128309).

Also, squares of exponentially odious numbers (A270428).

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 A270428.

Cf. A367801, A367803, A367804.

Cf. A000069, A128309, A270428, A354180.

Programs

Mathematica

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

PARI

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

Formula

a(n) = A270428(n)^2.

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=1} 1/p^A128309(k)) = Product_{p prime} f(1/p) = 1.62202332101829028287..., where f(x) = 1 + (2/(1-x^2) - Product_{k>=0} (1 - x^(2^k)) - Product_{k>=0} (1 - (-x)^(2^k)))/4.