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A133330
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Sums of exactly three positive octahedral numbers A005900.
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1
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3, 8, 13, 18, 21, 26, 31, 39, 44, 46, 51, 56, 57, 64, 69, 82, 87, 89, 92, 94, 97, 105, 107, 110, 123, 130, 132, 135, 148, 153, 158, 166, 171, 173, 176, 184, 189, 191, 196, 209, 214, 232, 233, 234, 237, 238, 243, 250, 251, 255, 256, 269, 275, 276, 281, 293, 294
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Pollock (1850) conjectured that every number is the sum of at most 7 octahedral numbers. If true, then every octahedral number is itself the sum of at most 7 octahedral numbers: do we have the sequences for those integers that are the sum of at most k octahedral numbers for for each k in the range 3<k<7 and the subsequences of the octahedral number values of those sequences? For example, the integers which are the sum of 3 positive octahedral numbers begin 3, 8, 13, 18, 21, 26, 31, 39, 44, 46; of those the first octahedral number is Oc(4) = Oc(2) + Oc(3) + Oc(3) = 6 + 19 + 19 = 44.
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REFERENCES
| Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Dover, 2005, cites the Pollock reference.
Pollock, F. "On the Extension of the Principle of Fermat's Theorem of the Polygonal Numbers to the Higher Orders of Series Whose Ultimate Differences Are Constant. With a New Theorem Proposed, Applicable to All the Orders." Abs. Papers Commun. Roy. Soc. London 5, 922-924, 1843-1850.
Agustin Moreno Canadas, On sums of figurate numbers by using techniques of poset representation theory, arXiv:0806.2486.
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MATHEMATICA
| lim = 300; oc[n_] := (2*n^3 + n)/3; nmax = Floor[Solve[oc[n] + oc[1] + oc[1] == lim, n][[1, 1, 2]]]; t = Table[ oc[n], {n, nmax}]; Select[ Union[ Flatten[ Outer[ Plus, t, t, t]]], # <= lim &] (* From Jean-François Alcover, Sep 08 2011 *)
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CROSSREFS
| Cf. A005900, A053676-A053678.
Sequence in context: A152676 A197062 A010064 * A190505 A184921 A095762
Adjacent sequences: A133327 A133328 A133329 * A133331 A133332 A133333
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KEYWORD
| easy,nonn
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AUTHOR
| Jonathan Vos Post (jvospost3(AT)gmail.com), Oct 18 2007
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