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A018227 Magic numbers: atoms with full shells containing any of these numbers of electrons are considered electronically stable. 20
2, 10, 18, 36, 54, 86, 118, 168, 218, 290, 362, 460, 558, 686, 814, 976, 1138, 1338, 1538, 1780, 2022, 2310, 2598, 2936, 3274, 3666, 4058, 4508, 4958, 5470, 5982, 6560, 7138, 7786, 8434, 9156, 9878, 10678, 11478, 12360, 13242, 14210, 15178 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Atomic numbers of noble elements in the periodic table.

Partial sums of A093907. - Lekraj Beedassy, Mar 24 2006

Comment from Don N. Page (don(AT)phys.ualberta.ca), Dec 12 2006: (Start)

"Relativistic corrections and instabilities to pair creation of electrons and positrons would occur even if one could have stable nuclei of arbitrarily many protons Z for the fixed value of the fine structure constant alpha ~ 1/137 in our universe.

"However, if one considered an imaginary universe with arbitrarily tiny alpha and a fixed point source of charge Z, one could have stable neutral atoms of Z nonrelativistic electrons of mass m for any Z, so long as one takes the limit Z alpha -> 0 by taking alpha -> 0 after fixing Z.

"One could then define noble elements to be given by the integer values of Z such that the ionization energy, in units of m c^2 alpha^2, of any such atom in its ground state with larger Z is less than that of the noble element (which appears to be the case for all the noble elements with the actual nonzero value of alpha).

"This sequence of idealized nonrelativistic noble elements with Z electrons would give an infinite sequence of integers Z, which may or may not be the same as that given by the explicit formula listed for the present sequence. It would likely be a difficult mathematical problem to calculate this infinite sequence." (End)

REFERENCES

A brief description is given under "Magic numbers" in the Encyclopedia Britannica.

S. Bjornholm, Clusters..., Contemp. Phys. 31 1990 pp. 309-324 (p. 312).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

D. Weise, The Pythagorean Approach to Problems of Periodicity in Science

D. Weise, Pythagorean Approach To Problems Of Periodicity In Fermionic System

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

FORMULA

a(n) = a(n-1) + ((2*n + 3 + (-1)^n)^2)/8; a(n) = (2*n^3 + 12*n^2 + 25*n - 6 + (-1)^n*(3*n + 6))/12 - Warut Roonguthai, Jun 20 2005

a(n) = n{(n+3)^2 + 5}/6 for even n, a(n) = n{(n+3)^2 + 2)/6 - 1 [or C(n+3,3) - 2, i.e., A000292(n) - 2] for odd n. - Lekraj Beedassy, Feb 02 2006

Partial sums of A116471. - Lekraj Beedassy, Mar 31 2006

From Daniel Forgues, May 01 2011 and May 02 2011: (Start)

a(n) = n*((n+3)^2 + 2)/6 + (n+2)*(1+(-1)^n)/4 - 1, n >= 1.

a(n) = (n+1)*(n+2)*(n+3)/6 + (n+2)*(1+(-1)^n)/4 - 2, n >= 1.

a(n) = T_{n+1} + (n+2)*(1+(-1)^n)/4 - 2, n >= 1,

where T_n is the n-th tetrahedral number.

G.f.: 2*x*(1 + 3*x - 2*x^2 - x^3 + x^4)/((1 - x)^4*(1 + x)^2). (End)

PROG

(MAGMA) [n*((n+3)^2 + 2)/6 + (n+2)*(1+(-1)^n)/4 - 1: n in [1..50]]; // Vincenzo Librandi, May 03 2011

CROSSREFS

Cf. A018226 for the magic numbers for nucleons (protons and neutrons).

Sequence in context: A282226 A082969 A173592 * A092062 A271786 A134251

Adjacent sequences:  A018224 A018225 A018226 * A018228 A018229 A018230

KEYWORD

nonn

AUTHOR

John Raithel (raithel(AT)rahul.net)

STATUS

approved

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Last modified January 17 22:49 EST 2019. Contains 319251 sequences. (Running on oeis4.)