A103981 -id:A103981 - cp's OEIS frontend

2, 5, 6, 9, 13, 17, 19, 21, 23, 31, 33, 53, 71, 87, 89, 93, 113, 123, 127, 157, 163, 167, 177, 181, 197, 201, 219, 229, 237, 321, 327, 347, 373, 393, 401, 409, 417, 419, 449, 487, 489, 503, 509, 519, 523, 537, 541, 563, 571, 577, 597, 599, 633, 647, 699, 751

Offset: 1

Jonathan Vos Post, Feb 23 2005

Because the cubic polynomial for octahedral numbers factors into n time a quadratic, the octahedral numbers can never be prime after a(3) = 19, but can be semiprime (if n is prime and 2*n^2+1 is triple a prime, or if n is triple a prime and 2*n^2+1 is prime). A005900(37) = 33781 = 11 * 37 * 83, three prime factors with same number of digits. A005900(41) = 45961 = 19 * 41 * 59, three prime factors with same number of digits. A005900(57) = 123481 = 19 * 67 * 97, three prime factors with same number of digits. A005900(67) = 200531 = 41 * 67 * 73, three prime factors with same number of digits. A005900(73) = 259369 = 11 * 17 * 19 * 73, four prime factors with same number of digits.

			93 is in this sequence because A005900(93) = (2*93^3 + 93)/3 = 536269 = 31 * 17299, which is semiprime.

J. H. Conway and R. K. Guy, The Book of Numbers, New York, Springer-Verlag, p. 50, 1996.
L. E. Dickson, History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, 1952.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
Jonathan Vos Post, Table of Polytope Numbers, Sorted, Through 1,000,000. [dead link]
Eric Weisstein's World of Mathematics, Octahedral Number.
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A001222, A005900, A103946, A103981.

Flatten[Position[Table[(2n^3+n)/3,{n,1000}],?(PrimeOmega[#]==2&)]] (* _Harvey P. Dale, Jun 17 2013 *)

isok(n) = bigomega((2*n^3+n)/3) == 2; \\ Michel Marcus, Dec 14 2015

Numbers k such that A005900(k) is a term of A001358.
Numbers k such that A103981(k) = 2.
Numbers k such that A001222(A005900(k)) = 2.
Numbers k such that Bigomega((2*k^3 + k)/3) = 2.

More terms from Harvey P. Dale, Jun 17 2013

cp's OEIS Frontend

A103982 Indices of octahedral numbers (A005900) which are semiprimes.

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