A033547 -id:A033547 - cp's OEIS frontend

A162626 If 0 <= n <= 3 then a(n) = n(n+1)(n+2)/3, if n >= 4 then a(n) = n(n^2+5)/3.

0, 2, 8, 20, 28, 50, 82, 126, 184, 258, 350, 462, 596, 754, 938, 1150, 1392, 1666, 1974, 2318, 2700, 3122, 3586, 4094, 4648, 5250, 5902, 6606, 7364, 8178, 9050, 9982, 10976, 12034, 13158, 14350, 15612, 16946, 18354, 19838, 21400, 23042, 24766, 26574

Offset: 0

Omar E. Pol, Jul 07 2009, Jul 13 2009

One way to generalize the magic number sequence in A018226.

A219239 Double magic numbers (in physics).

Original entry on oeis.org

4, 10, 16, 22, 28, 30, 36, 40, 48, 52, 56, 58, 70, 78, 84, 90, 100, 102, 110, 128, 132, 134, 146, 154, 164, 176, 208, 252

Offset: 1

Wolfdieter Lang, Dec 12 2012

For the magic numbers see A018226.
An atomic nucleus is called double magic if Z (number of protons in an atomic nucleus, atomic number) and N (number of neutrons) are both magic numbers. The nucleon or mass number (forget the Z electrons) is A = Z + N.
Each number a(n) is obtained in only one way as a sum of two (possibly equal) magic numbers. Only 28 is magic and double magic.

			Tin-132 is a double magic radionuclide (unstable isotope) with nucleon number A = 132 = a(21), Z = 50 and N = 82. Similarly for tin-100 with Z = N = 50. The stable primordial nuclide barium-132 is not double magic, because it has Z = 56 and N = 76.

Wikipedia, Double magic.
Wikipedia, List of Nuclides.

Cf. A018226, A033547.

a(n) is the sum of two numbers from [2, 8, 20, 28, 50, 82, 126] (the magic numbers A018226).

A230434 Magic numbers of nucleons. Another version of A018226, with 34 inserted.

Original entry on oeis.org

2, 8, 20, 28, 34, 50, 82, 126

Offset: 1

Omar E. Pol, Oct 22 2013

Union of 34 and A018226.

D. Steppenbeck et al., Evidence for a new nuclear 'magic number' from the level structure of 54Ca, Nature, 502 (2013), 207-210, DOI: 10.1038/nature12522.

Cf. A018226, A018227, A033547, A046939, A046940, A110856, A117612.

A369524 Array read by antidiagonals: T(n,k) is the number of length n necklaces using at most k colors with black beads always occurring in runs of even length.

Original entry on oeis.org

0, 1, 1, 2, 2, 0, 3, 4, 2, 1, 4, 7, 6, 3, 0, 5, 11, 14, 11, 3, 1, 6, 16, 28, 34, 18, 5, 0, 7, 22, 50, 87, 81, 38, 5, 1, 8, 29, 82, 191, 276, 227, 70, 8, 0, 9, 37, 126, 373, 759, 983, 615, 151, 10, 1, 10, 46, 184, 666, 1782, 3301, 3500, 1789, 314, 15, 0, 11, 56, 258, 1109, 3717, 9180, 14545, 13007, 5206, 684, 19, 1

Offset: 1

Maxim Karimov and Vladislav Sulima, Jan 25 2024

Equivalently, black beads can be considered to have length 2, while all other beads have length 1.

Column k is the "CIK" (necklace, indistinct, unlabeled) transform of {k-1, 1, 0, 0, 0, ...} (see C. Bower link). - Andrew Howroyd, Jan 25 2024

			n\k| 1  2   3     4      5       6       7        8         9 ...
---+-----------------------------------------------------------------
 1 | 0  1   2     3      4       5       6        7         8 ...A001477
 2 | 1  2   4     7     11      16      22       29        37 ...A000124
 3 | 0  2   6    14     28      50      82      126       184 ...A033547
 4 | 1  3  11    34     87     191     373      666      1109
 5 | 0  3  18    81    276     759    1782     3717      7080
 6 | 1  5  38   227    983    3301    9180    22163     47997
 7 | 0  5  70   615   3500   14545   48210   135155    333400
 8 | 1  8 151  1789  13007   66166  260113   844691   2370229
 9 | 0 10 314  5206  48820  304970 1423790  5358934  17110376
10 | 1 15 684 15490 186195 1425453 7897006 34438104 125093109
...

C. G. Bower, Transforms (2).

Columns 1..2 are A000035(n-1), A000358.
Rows 1..3 are A001477(k-1), A000124(k-1), A033547(k-1).
Cf. A000010 (phi), A075195 (all beads of same length).

function [res] = num2(n,k)
res=0;
for d=divisors(n)
    s=(k-1)^d;
    for i=1:floor(d/2)
        s=s + nchoosek(d-i-1,i-1) * d/i * (k-1)^(d-2*i);
    end
    res= res + eulerPhi(n/d) * s;
end
res=res/n;
end

T(n,k) = sum(d=1, n, eulerphi(d)*polcoef(log(1/(1 - (k-1)*x^d - x^(2*d)) + O(x*x^n)), n)/d)  \\ Andrew Howroyd, Jan 25 2024

T(n,k) = (1/n) * Sum_{d|n} phi(n/d) * ((k-1)^d + Sum_{i=1..floor(d/2)} binomial(d-i-1,i-1) * d/i * (k-1)^(d-2*i)), where phi(n) = A000010.

G.f. of column k: Sum_{d>=1} (phi(d)/d) * log(1/(1 - (k-1)*x^d - x^(2*d))). - Andrew Howroyd, Jan 25 2024

cp's OEIS Frontend

A162626 If 0 <= n <= 3 then a(n) = n(n+1)(n+2)/3, if n >= 4 then a(n) = n(n^2+5)/3.

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A219239 Double magic numbers (in physics).

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A230434 Magic numbers of nucleons. Another version of A018226, with 34 inserted.

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A369524 Array read by antidiagonals: T(n,k) is the number of length n necklaces using at most k colors with black beads always occurring in runs of even length.

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