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A367802
Exponentially odious squares.
5
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
1,2
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
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175 (2016), pp. 385-395.
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.
MATHEMATICA
odiousQ[n_] := OddQ[DigitCount[n, 2, 1]]; Select[Range[150]^2, AllTrue[FactorInteger[#][[;; , 2]], odiousQ] &]
PROG
(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);
CROSSREFS
Intersection of A000290 and A270428.
Sequence in context: A084617 A179126 A354180 * A369567 A340674 A068879
KEYWORD
nonn,easy,base
AUTHOR
Amiram Eldar, Dec 01 2023
STATUS
approved