

A303601


Number of ways to write n as a*(a+1)/2 + b*(b+1)/2 + Bell(k) + Bell(m) with 0 <= a <= b and 0 < k <= m, where Bell(k) denotes the kth Bell number A000110(k).


20



0, 1, 2, 3, 3, 4, 4, 5, 5, 6, 4, 5, 7, 5, 4, 7, 7, 7, 8, 8, 5, 9, 10, 7, 6, 9, 8, 8, 6, 7, 10, 10, 9, 8, 7, 8, 9, 10, 6, 9, 11, 7, 6, 8, 9, 10, 7, 10, 8, 7, 8, 10, 10, 9, 10, 8, 9, 13, 14, 10, 11, 12, 12, 9, 9, 12, 11, 13, 11, 9
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OFFSET

1,3


COMMENTS

Conjecture: a(n) > 0 for all n > 1. In other words, any integer n > 1 can be expressed as the sum of two triangular numbers and two Bell numbers.
This has been verified for all n = 2..7*10^8. Note that 111277 cannot be written as the sum of two squares and two Bell numbers.
As log(Bell(n)) is asymptotically equivalent to n*log(n), Bell numbers eventually grow faster than any exponential function.
See also A303389, A303540, A303543 and A303637 for similar conjectures.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei Sun, On universal sums of polygonal numbers, Sci. China Math. 58(2015), no. 7, 13671396.
ZhiWei Sun, Refining Lagrange's foursquare theorem, J. Number Theory 175(2017), 167190.
ZhiWei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97120.


EXAMPLE

a(2) = 1 with 2 = 0*(0+1)/2 + 0*(0+1)/2 + Bell(1) + Bell(1).
a(3) = 2 with 3 = 0*(0+1)/2 + 1*(1+1)/2 + Bell(1) + Bell(1) = 0*(0+1)/2 + 0*(0+1)/2 + Bell(1) + Bell(2).


MATHEMATICA

TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
b[n_]:=b[n]=BellB[n];
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n], i], 1], 4]==3&&Mod[Part[Part[f[n], i], 2], 2]==1], {i, 1, Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)(n>0&&g[n]);
tab={}; Do[r=0; k=1; Label[bb]; If[b[k]>n, Goto[aa]]; Do[If[QQ[4(nb[k]b[j])+1], Do[If[TQ[nb[k]b[j]x(x+1)/2], r=r+1], {x, 0, (Sqrt[4(nb[k]b[j])+1]1)/2}]], {j, 1, k}]; k=k+1; Goto[bb]; Label[aa];
tab=Append[tab, r], {n, 1, 70}]; Print[tab]


CROSSREFS

Cf. A000110, A000217, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303541, A303543, A303637.
Sequence in context: A279402 A324475 A189705 * A031247 A062575 A073188
Adjacent sequences: A303598 A303599 A303600 * A303602 A303603 A303604


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Apr 26 2018


STATUS

approved



