A367801 -id:A367801 - 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.

A367803 Exponentially evil squares.

Original entry on oeis.org

1, 64, 729, 1024, 4096, 15625, 46656, 59049, 117649, 262144, 531441, 746496, 1000000, 1048576, 1771561, 2985984, 3779136, 4826809, 7529536, 9765625, 11390625, 16000000, 16777216, 24137569, 34012224, 47045881, 60466176, 64000000, 85766121, 113379904, 120472576, 148035889

Offset: 1

Amiram Eldar, Dec 01 2023

Numbers whose prime factorization contains only exponents that are even evil numbers (A125592).
Also, squares of exponentially evil numbers (A262675).
Also, numbers with an equal number of exponentially odious and exponentially evil divisors, i.e., numbers k such that A366901(k) = A366902(k). - Amiram Eldar, Feb 26 2024

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 A262675.
Cf. A366901, A366902, A367801, A367802, A367804.
Cf. A001969, A125592.

evilQ[n_] := EvenQ[DigitCount[n, 2, 1]]; Select[Range[10^4]^2, #== 1 || AllTrue[FactorInteger[#][[;;, 2]], evilQ] &]

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

a(n) = A262675(n)^2.

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=1} 1/p^A125592(k)) = Product_{p prime} f(1/p) = 1.01833932269003592136..., where f(x) = (2/(1-x^2) + Product_{k>=0} (1 - x^(2^k)) + Product_{k>=0} (1 - (-x)^(2^k)))/4.

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

Original entry on oeis.org

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

Amiram Eldar, Dec 01 2023

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

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 A262675 and A268335.
Cf. A367801, A367802, A367803.
Cf. A129771.

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

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

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.

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A367802 Exponentially odious squares.

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A367803 Exponentially evil squares.

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

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