A160914 -id:A160914 - cp's OEIS frontend

A167268 Janet's sequence: Number of elements for each successively filled electronic subshell of an atom.

2, 2, 6, 2, 6, 2, 10, 6, 2, 10, 6, 2, 14, 10, 6, 2, 14, 10, 6, 2, 18, 14, 10, 6, 2, 18, 14, 10, 6, 2, 22, 18, 14, 10, 6, 2, 22, 18, 14, 10, 6, 2, 26, 22, 18, 14, 10, 6, 2, 26, 22, 18, 14, 10, 6, 2, 30, 26, 22, 18, 14, 10, 6, 2, 30, 26, 22, 18, 14, 10, 6

Offset: 1

Paul Curtz, Oct 31 2009

The s-block is placed at
* the left, except row 1 (for Mendeleev-Moseley-Seaborg periodic table)
* the right (for Janet periodic table)
The number of elements in each block of Janet's periodic table are
  s-block (1s to 8s): 8*2  = 16 elements;
  p-block (2p to 7p): 6*6  = 36 elements;
  d-block (3d to 6d): 4*10 = 40 elements;
  f-block (4f to 5f): 2*14 = 28 elements.
The atomic numbers of elements in each block of Janet's periodic table are
s-block: 1,2, 3,4, 11,12, 19,20, 37,38, 55,56, 87,88, 119,120 (cf. A160914)
p-block: 5..10, 13..18, 31..36, 49..54, 81..86, 113..118 (cf. A138469)
d-block: 21..30, 39..48, 71..80, 103..112 (cf. A199934)
f-block: 57..70, 89..102 (cf. A217923).
The number of elements for each successive filled subshell are
  s-block (1s): a(0)=2 terms (for H and He, i.e., 1 and 2);
  s-block (2s): a(1)=2 terms (for Li and Be, i.e., 3 and 4);
  p-block (2p): a(2)=6 terms (for B,C,N,O,F,Ne, i.e., 5 to 10);
  s-block (3s): a(3)=2 terms (for Na,Mg, i.e., 11 and 12);
  p-block (3p): a(4)=6 terms (for Al,Si,P,S,Cl,Ar, i.e., 13 to 18);
  ...
Reference, 2 leaflet 2, with Janet form (5). Extended.
From Daniel Forgues, May 09 2011: (Start)
Janet also envisaged an 'element zero' ('neutronium'?) - whose 'atom' would consist of two neutrons (and thus zero electrons, which would give a(0) = 0) and he speculated that this would be the link to a mirror-image table of elements with negative atomic numbers - in effect anti-matter (which would give a(-n) = -a(n), since positrons are negated electrons).
Maximum number of electrons for successive subshells of each shell of an atom, in the building up order (per aufbau principle and Madelung's rule).
Every term is twice an odd number since each filled subshell in block l has m going from -l to + l (2l+1 values,) each with 2 electrons (spin +1/2 and spin -1/2).
Blocks:
  l=0: s (2 electrons) (first subshell of a shell, new period of Mendeleev's table)
  l=1: p (6 electrons) (except for first shell, last subshell of a shell)
  l=2: d (10 electrons)
  l=3: f (14 electrons)
  l=4: g (18 electrons)
  ...
  l=l: ... (2*(2l+1) electrons)
The first subshell of the k-th shell has l = 0, k >= 1.
The second subshell of the k-th shell has l = floor(k/2), k >= 2.
The following subshells of the k-th shell have l decrementing down to 1, k >= 2.
(End)
Concatenation of finite arithmetic sequences, each followed by 2: { }, 2, { }, 2, {6}, 2, {6}, 2, {10, 6}, 2, {10, 6}, 2, {14, 10, 6}, 2, {14, 10, 6}, 2, {18, 14, 10, 6}, 2, {18, 14, 10, 6}, 2, ... - Daniel Forgues, May 15 2011
Named after the French mathematician Charles Janet (1849-1932). - Amiram Eldar, Jun 22 2021

Charles Janet, Considérations sur la structure du noyau de l'atome, N 5, Décembre 1929, Beauvais, 2+45 pages, 4 leaflets.

Daniel Forgues, Table of n, a(n) for n = 1..9899
The Chemogenesis web book, Quantum Numbers to Periodic Tables.
Szymon Łukaszyk and Wawrzyniec Bieniawski, Assembly Theory of Binary Messages (How to Assemble a Black Hole and Use it to Assemble New Binary Information?), Preprints (2024).

Cf. A016825, A160914, A138469, A199934, A216607.

row[n_] := Range[2, 4*Floor[n/2] + 2, 4]; Flatten[ Table[ row[n] // Reverse, {n, 0, 15}]] (* Jean-François Alcover, Oct 02 2012 *)

from mpmath.libmp import sqrtrem
def a(n):
    s, r = sqrtrem(n)
    return 4 * (-n % (s + (r>s))) + 2
# Christoph B. Kassir, Apr 07 2022

a(n) = 4*((-n) mod round(sqrt(n))) + 2. - Jon E. Schoenfield, Sep 08 2013

a(n) = 4*A216607(n) + 2. - Szymon Lukaszyk, Oct 27 2023

Edited by Daniel Forgues, May 09 2011

A138469 Atomic numbers of p-block elements.

Original entry on oeis.org

5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 18, 31, 32, 33, 34, 35, 36, 49, 50, 51, 52, 53, 54, 81, 82, 83, 84, 85, 86, 113, 114, 115, 116, 117, 118

Offset: 1

Paul Curtz, May 09 2008

6 X 6 square of orbital p filling elements in Mendeleyev-Seaborg or Janet-Tarantola periodic table.
The sequence is considered full if we assume that all elements with 8 shells or more, i.e. of periods 8 and higher in Mendeleev's periodic table, have extremely unstable nuclei (extremely short half-life.)
The atomic numbers of elements in each block are
  s-block: 1,2, 3,4, 11,12, 19,20, 37,38, 55,56, 87,88 for Mendeleev's periodic table (Cf. A160914)
  s-block: 1,2, 3,4, 11,12, 19,20, 37,38, 55,56, 87,88, 119,120 for Janet's periodic table (Cf. A160914)
  p-block: 5..10, 13..18, 31..36, 49..54, 81..86, 113..118 (Cf. A138469)
  d-block: 21..30, 39..48, 71..80, 103..112 (Cf. A199934)
  f-block: 57..70, 89..102 (Cf. A217923)

OEIS Wiki, Periodic table

Cf. A130517, A131603, A167268.
Cf. A160914 Atomic numbers of s-block elements.
Cf. A138469 Atomic numbers of p-block elements.
Cf. A199934 Atomic numbers of d-block elements.
Cf. A217923 Atomic numbers of f-block elements.

Edited by Daniel Forgues, May 15 2011

Ref. to A160914 and A199934 inserted by Jean-François Alcover, Oct 15 2012

A199934 Extended d-block elements for Janet table.

Original entry on oeis.org

21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 347

Offset: 1

Paul Curtz, Nov 12 2011

After the s-block (A160914) and the p-block (A138469), we present the d-block (4*10) elements for the Janet table, in A167268. Janet published his table in a 1928 book (see below), 24 of planche 7, in which he also introduced the s, p, d, f blocks (called nappes 1, 2, 3, 4).
a(n) also comes from one of the two classical Mendeleyev-Moseley-Seaborg periodic table(s) i.e. (see A138096):
                                                2
 4                                       5 to  10
12                                      13 to  18
20                          21 to  30   31 to  36
38                          39 to  48   49 to  54
56  57 to  70   71 to  80   81 to  86
88  89 to 102  103 to 112  113 to 118.
(The second is in A134982).

Charles Janet, Essais de classification hélicoidale des éléments chimiques, April 1928, N3, Beauvais, 2 + 104 pages, 4 leaflets (4 to 7).

Wikipedia, The Janet periodic table

A217923 F-block elements for Janet periodic table.

Original entry on oeis.org

57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102

Offset: 1

Jean-François Alcover, Oct 15 2012

From Paul Curtz, Oct 26 2012: (Start)
Tarantola's formulas for the Janet table:
Let D(n) = n*(n+1)*(n+2)/6 + (1-(-1)^n)*(n+1)/4 = A168380(n).
The row R at which the element with atomic number Z is to be placed is the unique value of R satisfying D(R-1) < Z <= D(R).
Once the row number R is determined, the column number is, from right to left, C = D(R) - Z + 1.
Example: Z=109. D(7) < Z <= D(8) = 120. C = 120 - 109 + 1 = 12. (End)

Wikipedia, The Janet periodic table

Cf. A160914 (s-block), A138469 (p-block), A199934 (d-block).

Mathematica
```
Join[Range[57,70], Range[89,102]]
```

cp's OEIS Frontend

A167268 Janet's sequence: Number of elements for each successively filled electronic subshell of an atom.

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A138469 Atomic numbers of p-block elements.

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A199934 Extended d-block elements for Janet table.

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A217923 F-block elements for Janet periodic table.

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