

A257162


Number of ordered ways to write n as floor(x/5) + floor(y/6) + floor(z/7), where x is a pentagonal number, y is a hexagonal number and z is a heptagonal number.


1



8, 12, 18, 13, 21, 15, 16, 18, 21, 18, 13, 27, 20, 18, 19, 26, 22, 19, 30, 21, 22, 25, 31, 25, 20, 34, 23, 31, 21, 35, 24, 33, 28, 30, 30, 32, 34, 27, 33, 28, 35, 33, 38, 30, 31, 32, 37, 30, 34, 39, 42, 35, 32, 31, 39, 33, 40, 38, 41, 36, 41, 36, 37, 32, 41, 43, 34, 42, 43, 42, 37
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OFFSET

0,1


COMMENTS

Conjecture: a(n) > 0 for all n. Moreover, for any i,j,k = 3,4,5,... all sufficiently large integers can be written as floor(x/i) + floor(y/j) + floor(z/k), where x is an igonal number, y is a jgonal number and z is a kgonal number.
Note that if {i,j,k} is a subset of {18,19} then 48 cannot be written as floor(x/i) + floor(y/j) + floor(z/k), where x is an igonal number, y is a jgonal number and z is a kgonal number.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 0..3000
ZhiWei Sun, Natural numbers represented by floor(x^2/a)+floor(y^2/b)+floor(z^2/c), arXiv:1504.01608 [math.NT], 2015.


EXAMPLE

a(0) = 8 since 0 = floor(x/5) + floor(y/6) + floor(z/7) for any x,y,z in {0,1}, and 0 and 1 are the only mgonal numbers smaller than m.


MATHEMATICA

SQ[n_]:=IntegerQ[Sqrt[n]]
p[m_, n_]:=p[m, n]=(m2)*Binomial[n, 2]+n
PQ[m_, n_]:=SQ[8(m2)n+(m4)^2]&&(n==0Mod[Sqrt[8(m2)n+(m4)^2]+m4, 2m4]==0)
Do[a=0; Do[If[PQ[7, 7(nFloor[p[5, x]/5]Floor[p[6, y]/6])+r], a=a+1], {x, 0, (Sqrt[24(5*n+4)+1]+1)/6}, {y, 0, (Sqrt[8*(6*(nFloor[p[5, x]/5])+5)+1]+1)/4}, {r, 0, 6}];
Print[n, " ", a]; Continue, {n, 0, 70}]


CROSSREFS

Cf. A000326, A000384, A000566.
Sequence in context: A175975 A030752 A091523 * A352880 A104201 A036327
Adjacent sequences: A257159 A257160 A257161 * A257163 A257164 A257165


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Apr 16 2015


STATUS

approved



