Szymon Lukaszyk - cp's OEIS frontend

A372152 Number of k in the range 2^n <= k < 2^(n+1) whose shortest addition chain does not have length n, n+1 or n+2.

0, 0, 0, 0, 2, 9, 30, 80, 193, 432, 925, 1928, 3953, 8024, 16189, 32544

Offset: 0

Szymon Lukaszyk, Apr 20 2024

The length of the shortest addition chain for k is A003313(k).
Dividing natural numbers into sections 2^n <= k < 2^(n+1), some of the 2^n numbers available in a section have the shortest addition chains given by
 n (for k=2^n),
 n+1 (for k=2^n+2^m, m in [0..n-1], A048645), or
 n+2 (for some k in A072823).
The sequence gives the numbers of k within each section (N_oth) that have the shortest addition chains other than n, n+1, and n+2.
In particular for 4 <= n <= 6, N_oth = 2^n - n^2 + 2 and for n >= 7, N_oth = 2^n - n^2 + 1.

S. Łukaszyk and W. Bieniawski, Assembly Theory of Binary Messages, Mathematics, 12(10) (2024), 1600.

Cf. A003313, A048645, A072823, A014701, A024012.

A371894 Integers k such that the number of multiplications to compute the k-th power by the Chandah-sutra method (A014701) is larger than the length of the shortest addition chain for k (A003313).

Original entry on oeis.org

15, 23, 27, 30, 31, 39, 43, 45, 46, 47, 51, 54, 55, 59, 60, 61, 62, 63, 75, 77, 78, 79, 83, 85, 86, 87, 90, 91, 92, 93, 94, 95, 99, 102, 103, 107, 108, 109, 110, 111, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 135, 143, 147, 149, 150, 151, 153, 154, 155

Offset: 1

Szymon Lukaszyk, Apr 11 2024

nonn

All terms have a binary weight A000120(k) >= 4.

Szymon Lukaszyk, Table of n, a(n) for n = 1..8876

Cf. A003313, A014701, A000120.

A349078 List of positive integer triples a, b, c with the property a^2 + b^2 = c^4 and a

Original entry on oeis.org

7, 24, 5, 15, 20, 5, 28, 96, 10, 60, 80, 10, 65, 156, 13, 119, 120, 13, 63, 216, 15, 135, 180, 15, 136, 255, 17, 161, 240, 17, 112, 384, 20, 240, 320, 20, 175, 600, 25, 220, 585, 25, 336, 527, 25, 375, 500, 25, 260, 624, 26, 476, 480, 26, 41, 840, 29, 580, 609, 29

Offset: 1

Szymon Lukaszyk, Dec 30 2021

For a given c, the triples are sorted in order of increasing a.

Triples are listed in order of nondecreasing c. - Chai Wah Wu, Jan 02 2022

Chai Wah Wu, Table of n, a(n) for n = 1..10062

Cf. A009000.

A338992 Number of vertices of a hexagonal tessellation that lie on subsequent circles centered at the center of one hexagon.

Original entry on oeis.org

0, 6, 6, 12, 12, 6, 12, 6, 12, 12, 12, 12, 18, 12, 12, 6, 12, 12, 12, 12, 24, 12, 6, 12, 12, 12, 6, 12, 12, 24, 12, 12, 12, 12, 12, 18, 12, 12, 12, 12, 18, 12, 12, 12, 24, 12, 12, 12, 12, 24, 6, 24, 12, 12, 12, 12, 6, 12, 24, 12, 12, 12, 12, 12, 12, 12, 24, 12, 18

Offset: 0

Szymon Lukaszyk, Nov 17 2020

nonn

Radii of these circles are square roots of A202822.

This is A217219 with zeros dropped, except for a(0). - Andrey Zabolotskiy, Jun 21 2022

Szymon Lukaszyk, MATLAB script
Szymon Lukaszyk, A short note about the geometry of graphene, ResearchGate, (2020).
Szymon Lukaszyk, Illustration of the subsequent 36 radii around a hex center.

Cf. A202822, A217219, A338947 (similar but with circles centered at a vertex of one hexagon).

A338947 Number of vertices of a hexagonal tessellation that lie on subsequent circles centered at a vertex of one hexagon.

Original entry on oeis.org

1, 3, 6, 3, 6, 6, 6, 6, 3, 6, 12, 3, 6, 6, 6, 6, 6, 12, 6, 6, 9, 6, 12, 6, 12, 3, 6, 6, 6, 6, 6, 6, 12, 12, 12, 6, 3, 6, 6, 6, 12, 6, 12, 3, 6, 6, 12, 12, 6, 6, 18, 6, 6, 12, 6, 6, 9, 12, 6, 6, 6, 12, 12, 6, 6, 9, 6, 12, 6, 6, 12, 12, 6, 6, 12, 6, 12, 6, 6, 6, 12, 12, 3, 12, 6, 6, 24, 6, 12, 6, 3, 12, 6, 6, 12, 6, 6, 12

Offset: 0

Szymon Lukaszyk, Nov 17 2020

nonn

Radii of these circles are square roots of A003136.

This is A113062 with zeros dropped. - Andrey Zabolotskiy, Jun 21 2022

Szymon Lukaszyk, A short note about the geometry of graphene, ResearchGate, (2020).
Szymon Lukaszyk, Illustration of the subsequent 51 radii around a hexagon vertex.
S. Reich et al., Tight-binding description of graphene (up to third-nearest neighbors circles).

Cf. A003136, A113062; A338992 (similar but with circles centered at the center of one hexagon); A104888, A106030, A317990 (terms multiplied by 3 in agreement for up to 17 term).

a(0) changed from 0 to 1 by Andrey Zabolotskiy, Jun 21 2022

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User: Szymon Lukaszyk

A372152 Number of k in the range 2^n <= k < 2^(n+1) whose shortest addition chain does not have length n, n+1 or n+2.

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A371894 Integers k such that the number of multiplications to compute the k-th power by the Chandah-sutra method (A014701) is larger than the length of the shortest addition chain for k (A003313).

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A349078 List of positive integer triples a, b, c with the property a^2 + b^2 = c^4 and a

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A338992 Number of vertices of a hexagonal tessellation that lie on subsequent circles centered at the center of one hexagon.

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A338947 Number of vertices of a hexagonal tessellation that lie on subsequent circles centered at a vertex of one hexagon.

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