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A280386 Number of ways to write n as the sum of two squares and a term of A000009. 3
1, 2, 3, 3, 4, 5, 4, 4, 4, 6, 5, 6, 5, 6, 5, 5, 5, 6, 7, 7, 6, 7, 7, 5, 4, 7, 7, 9, 5, 7, 8, 7, 6, 5, 9, 6, 8, 8, 6, 10, 6, 9, 7, 8, 5, 7, 10, 7, 5, 6, 9, 9, 7, 10, 11, 10, 6, 9, 8, 5, 5, 8, 10, 10, 6, 8, 10, 10, 7, 8, 9, 10, 8, 8, 8, 9, 10, 7, 8, 11 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Conjecture: a(n) > 0 for all n > 0.
Note that the main term of log A000009(n) is Pi*sqrt(n/3). So, A000009(n) eventually grows faster than any polynomial.
The conjecture was verified by the author for n up to 4*10^6. After learning this conjecture from the author, Prof. Qing-Hu Hou at Tianjin Univ. finished his verification of the above conjecture for n up to 10^9. - Zhi-Wei Sun, Jan 02 2017
LINKS
Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28--Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.
EXAMPLE
a(1) = 1 since 1 = 0^2 + 0^2 + 1 with 1 = A000009(1) = A000009(2).
a(2) = 2 since 2 = 0^2 + 1^2 + 1 = 0^2 + 0^2 + 2 with 1 = A000009(1) = A000009(2) and 2 = A000009(3) = A000009(4).
MATHEMATICA
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
q[n_]:=q[n]=PartitionsQ[n];
ex={}; Do[r=0; m=2; Label[bb]; If[q[m]>n, Goto[cc]]; Do[If[SQ[n-q[m]-x^2], r=r+1], {x, 0, Sqrt[(n-q[m])/2]}]; m=m+If[m<3, 2, 1]; Goto[bb]; Label[cc]; ex=Append[ex, r]; Continue, {n, 1, 80}]
CROSSREFS
Sequence in context: A325784 A244929 A302920 * A204979 A243351 A071585
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 01 2017
STATUS
approved

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Last modified March 29 09:44 EDT 2024. Contains 371268 sequences. (Running on oeis4.)