A372099 -id:A372099 - cp's OEIS frontend

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

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.

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

Original entry on oeis.org

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.

cp's OEIS Frontend

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

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

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PARI

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

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