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A103982 Indices of octahedral numbers (A005900) which are semiprimes. 2
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 (list; graph; refs; listen; history; text; internal format)
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
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
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
Sequence in context: A104857 A285899 A055198 * A030488 A190678 A255737
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Feb 23 2005
EXTENSIONS
More terms from Harvey P. Dale, Jun 17 2013
STATUS
approved

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Last modified February 21 06:16 EST 2024. Contains 370219 sequences. (Running on oeis4.)