A078839 -id:A078839 - cp's OEIS frontend

A372097 Exponents k where A000120(3^k) - A070939(3^k)/2 reaches a new minimum.

0, 2, 4, 7, 16, 24, 40, 49, 53, 102, 104, 126, 174, 226, 379, 768, 831, 832, 1439, 1452, 1914, 2291, 2731, 3000, 3363, 3472, 5608, 5883, 6725, 6787, 7438, 8786, 10280, 11948, 12190, 13135, 15170, 15645, 22407, 26232, 27099, 32773, 33085, 40189, 40523, 48068, 51187

Offset: 1

Hugo Pfoertner, Apr 25 2024

nonn

These are the k-values of the lower envelope of the scatter band of the deviation of the binary weight of 3^k from half the length of the corresponding binary number. The corresponding negated differences are given in A372098.

Hugo Pfoertner, Table of n, a(n) for n = 1..99
Hugo Pfoertner, Illustration of scatter band bounded by lower and upper records, up to exponents k=8*10^6.

Cf. A000120, A000244, A011754, A070939, A078839, A372098, A372099, A372100.

a372097(upto) = {my (dm=-oo); for (k=0, upto, my (p=3^k, h=hammingweight(p), b=#binary(p)/2,d=b-h); if (d>dm, print1(k,", "); dm=d))};
a372097(60000)

A372098 a(n) = A070939(3^k) - 2*A000120(3^k) with k = A372097(n).

Original entry on oeis.org

-1, 0, 1, 2, 4, 7, 8, 12, 15, 18, 25, 26, 30, 51, 75, 78, 84, 129, 133, 148, 170, 180, 183, 189, 209, 265, 279, 285, 287, 336, 369, 388, 406, 412, 445, 469, 496, 581, 711, 737, 741, 742, 873, 939, 994, 1044, 1078, 1111, 1157, 1158, 1492, 1636, 1767, 1914, 1933

Offset: 1

Hugo Pfoertner, Apr 25 2024

sign

a(n)/2 are the negated differences at supporting points of the lower envelope of the scatter band of the deviation of the binary weight of 3^k from half the length of the corresponding binary number.

Hugo Pfoertner, Table of n, a(n) for n = 1..99

Cf. A000120, A000244, A011754, A070939, A078839, A372097, A372099, A372100.

A372099 Exponents k where A000120(3^k) - A070939(3^k)/2 reaches a new maximum.

Original entry on oeis.org

0, 1, 3, 5, 11, 27, 71, 119, 140, 158, 198, 218, 441, 537, 538, 868, 1092, 2128, 2294, 2343, 2811, 2911, 3849, 4003, 4655, 5079, 5279, 5920, 6269, 6603, 10181, 10574, 12801, 12803, 15563, 15784, 16054, 16253, 17127, 18257, 20187, 21934, 34633, 49209, 76791, 78938

Offset: 1

Hugo Pfoertner, Apr 25 2024

nonn

These are the k-values of the upper envelope of the scatter band of the deviation of the binary weight of 3^k from half the length of the corresponding binary number. The corresponding differences are given in A372100.

Hugo Pfoertner, Table of n, a(n) for n = 1..96
Hugo Pfoertner, Illustration of scatter band bounded by lower and upper records, up to exponents k=8.5*10^6.

Cf. A000120, A000244, A011754, A070939, A078839, A372097, A372098, A372100.

a372099(upto) = {my(dm=oo); for (k=0, upto, my (p=3^k, h=hammingweight(p), b=#binary(p)/2, d=b-h); if (d

A372100 a(n) = 2*A000120(3^k) - A070939(3^k) with k = A372099(n).

Original entry on oeis.org

1, 2, 3, 4, 8, 17, 23, 29, 38, 39, 44, 56, 57, 58, 91, 114, 145, 147, 156, 168, 182, 208, 219, 239, 277, 297, 300, 307, 331, 360, 367, 442, 452, 477, 487, 492, 507, 513, 568, 571, 614, 893, 963, 1275, 1283, 1288, 1440, 1563, 1702, 1957, 2019, 2440, 2471, 2566, 3004

Offset: 1

Hugo Pfoertner, Apr 25 2024

nonn

a(n)/2 are the differences at supporting points of the upper envelope of the scatter band of the deviation of the binary weight of 3^k from half the length of the corresponding binary number.

Hugo Pfoertner, Table of n, a(n) for n = 1..96

Cf. A000120, A000244, A011754, A070939, A078839, A372097, A372098, A372099.

A371970 Exponents k such that the binary expansion of 3^k has an even number of ones.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 12, 14, 17, 18, 21, 23, 24, 25, 26, 27, 31, 32, 33, 35, 37, 38, 39, 40, 42, 44, 45, 47, 51, 52, 55, 57, 58, 59, 60, 61, 64, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 96, 99, 102, 104, 105, 106, 109, 112, 116, 127, 131, 132, 133, 134, 135, 136

Offset: 1

Hugo Pfoertner, Apr 24 2024

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Complement of A223024.

Cf. A000120, A000244, A001969, A011754, A078839.

q:= n-> is(add(i, i=Bits[Split](3^n))::even):
select(q, [$0..150])[];  # Alois P. Heinz, Apr 24 2024

Select[Range[136], EvenQ@ DigitCount[3^#, 2, 1] &] (* Michael De Vlieger, Apr 24 2024 *)

is_a371970(k) = hammingweight(3^k)%2 == 0

cp's OEIS Frontend

A372097 Exponents k where A000120(3^k) - A070939(3^k)/2 reaches a new minimum.

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A372098 a(n) = A070939(3^k) - 2*A000120(3^k) with k = A372097(n).

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A372099 Exponents k where A000120(3^k) - A070939(3^k)/2 reaches a new maximum.

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PARI

A372100 a(n) = 2*A000120(3^k) - A070939(3^k) with k = A372099(n).

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A371970 Exponents k such that the binary expansion of 3^k has an even number of ones.

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