A354180 -id:A354180 - cp's OEIS frontend

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

Amiram Eldar, Dec 01 2023

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).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175 (2016), pp. 385-395.

Intersection of A000290 and A270428.
Cf. A367801, A367803, A367804.
Cf. A000069, A128309, A270428, A354180.

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

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);

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.

A369567 Powerful exponentially 2^n-numbers: numbers whose prime factorization contains only exponents that are powers of 2 that are larger than 1.

Original entry on oeis.org

Offset: 1

Amiram Eldar, Jan 26 2024

nonn

First Differs from A354180 and A367802 at n = 113.
Also, exponentially 2^n-numbers that are squares.
Also, squares of exponentially 2^n-numbers.

Intersection of A001694 and A138302.
Intersection of A000290 and A138302.
Cf. A354180, A367802.

q[n_] := AllTrue[FactorInteger[n][[;; , 2]], # > 1 && # == 2^IntegerExponent[#, 2] &]; Select[Range[3300], # == 1 || q[#] &]

is(n) = {my(e = factor(n)[, 2]); if(n == 1, 1, for(i = 1, #e, if(e[i] == 1 || e[i] >> valuation(e[i], 2) > 1, return(0))); 1);}

a(n) = A138302(n)^2.

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=1} 1/p^(2^k)) = 1.62194750148969761827... .

cp's OEIS Frontend

A367802 Exponentially odious squares.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula