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

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

1,1

Comments

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.

References

Conway, J. H. and Guy, R. K. The Book of Numbers. New York, Springer-Verlag, p. 50, 1996

Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, 1952.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.

J. V. Post, Table of Polytope Numbers, Sorted, Through 1,000,000.

Eric Weisstein's World of Mathematics, Semiprime.

Eric Weisstein's World of Mathematics, Octahedral Number

Formula

n such that A005900(n) is an element of A001358. n such that A103981(n) = 2. n such that A001222(A005900(n)) = 2. n such that Bigomega((2*n^3 + n)/3) = 2.

Example

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

Mathematica

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

Prog

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

Crossrefs

Cf. A001222, A005900, A103946, A103981.

Sequence in context: A104857 A285899 A055198 * A030488 A190678 A255737

Adjacent sequences: A103979 A103980 A103981 * A103983 A103984 A103985

Keyword

easy,nonn

Author

Jonathan Vos Post, Feb 23 2005

Extensions

More terms from Harvey P. Dale, Jun 17 2013

Status

approved