A143928 2*p^2, for p an odd prime.

18, 50, 98, 242, 338, 578, 722, 1058, 1682, 1922, 2738, 3362, 3698, 4418, 5618, 6962, 7442, 8978, 10082, 10658, 12482, 13778, 15842, 18818, 20402, 21218, 22898, 23762, 25538, 32258, 34322, 37538, 38642, 44402, 45602, 49298, 53138, 55778, 59858

Offset: 1

Author

Jonathan Vos Post, Sep 05 2008

Keywords

• easy
• nonn

Comments

For these numbers m, there are precisely 5 groups of order m, hence it is a subsequence of A054397. The 5 groups are C_{2*p^2}, C_2 X (C_p X C_p), C_p^2 : C_2 ~ D_{2*p^2}, and two non-isomorphic groups (C_p X C_p) : C_2, where C, D mean cyclic, dihedral groups of the stated order; the symbols ~, X and : mean isomorphic to, direct and semidirect products respectively. - Bernard Schott, Dec 10 2021

Examples

			a(1) = 2*A065091(1)^2 = 2*3^2 = 18.
a(2) = 2*A065091(2)^2 = 2*5^2 = 50.
a(3) = 2*A065091(3)^2 = 2*7^2 = 98.
		

References

• Pascal Ortiz, Exercices d'Algèbre, Collection CAPES / Agrégation, Ellipses, problème 1.35, pp. 70-74, 2004.

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Michael Hilgemann and Siu-Hung Ng, Hopf algebras of dimension 2p^2, arXiv:0809.0699 [math.QA], 2008.

Crossrefs

Subsequence of A079704.

Cf. A054397, A065091.

Programs

Mathematica

2#^2&/@Prime[Range[2,40]] (* Harvey P. Dale, Jul 23 2021 *)

Python

from sympy import prime
def a(n): return 2*prime(n+1)**2
print([a(n) for n in range(1, 40)]) # Michael S. Branicky, Dec 10 2021

Formula

a(n) = A079704(n+1) for n>0.