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A061336
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Smallest number of triangular numbers which sum to n.
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17
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0, 1, 2, 1, 2, 3, 1, 2, 3, 2, 1, 2, 2, 2, 3, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2, 2, 3, 2, 1, 2, 2, 2, 3, 3, 2, 3, 1, 2, 2, 2, 3, 3, 2, 2, 3, 1, 2, 3, 2, 2, 3, 2, 3, 3, 3, 1, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 1, 2, 3, 2, 2, 3, 2, 2, 3, 3, 2, 3, 1, 2, 3, 2, 3, 2, 2, 3, 3, 2, 2, 3, 2, 1, 2, 2, 2, 3, 3, 2, 3, 2, 2, 2, 2, 3, 3
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OFFSET
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0,3
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COMMENTS
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a(n)=3 if n=5 or 8 mod 9, since triangular numbers are {0,1,3,6} mod 9.
In September 1636, Fermat, in a letter to Mersenne, made the statement that every number is a sum of at most three triangular numbers. This was proved by Gauss, who noted this event in his diary on July 10 1796 with the notation:
EYPHKA! num = DELTA + DELTA + DELTA (where Y is in fact the Greek letter Upsilon and DELTA is the Greek letter of that name).
This proof was published in his book Disquisitiones Arithmeticae, Leipzig, 1801. (End)
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REFERENCES
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Elena Deza and Michel Marie Deza, Fermat's polygonal number theorem, Figurate numbers, World Scientific Publishing (2012), Chapter 5, pp. 313-377.
C. F. Gauss, Disquisitiones Arithmeticae, Yale University Press, 1966, New Haven and London, p. 342, art. 293.
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LINKS
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FORMULA
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a(n) = 0 if n=0, otherwise 1 if n is in A000217, otherwise 2 if n is in A051533, otherwise 3 in which case n is in A020757.
a(n) <= 3 (proposed by Fermat and proved by Gauss). - Bernard Schott, Jul 16 2022
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EXAMPLE
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a(3)=1 since 3=3, a(4)=2 since 4=1+3, a(5)=3 since 5=1+1+3, with 1 and 3 being triangular.
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MATHEMATICA
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t[n_]:=n*(n+1)/2; a[0]=0; a[n_]:=Block[ {k=1, tt= t/@ Range[Sqrt[2*n]]}, Off[IntegerPartitions::take]; While[{} == IntegerPartitions[n, {k}, tt, 1], k++]; k]; a/@ Range[0, 104] (* Giovanni Resta, Jun 09 2015 *)
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PROG
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(PARI) see A283370 for generic code, working but not optimized for this case of triangular numbers. - M. F. Hasler, Mar 06 2017
(PARI) a(n)=my(m=n%9, f); if(m==5 || m==8, return(3)); f=factor(4*n+1); for(i=1, #f~, if(f[i, 2]%2 && f[i, 1]%4==3, return(3))); if(ispolygonal(n, 3), n>0, 2) \\ Charles R Greathouse IV, Mar 17 2022
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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