A367802 -id:A367802 - cp's OEIS frontend

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

Amiram Eldar, Dec 01 2023

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.

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.

Intersection of A268335 and A270428.
Cf. A367802, A367803, A367804.
Subsequences: A005117, A092759.
Cf. A092246.
Cf. A077377, A348506.

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

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

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.

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

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, 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

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

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

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