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A322366 Number of integers k in {0,1,...,n} such that k identical test tubes can be balanced in a centrifuge with n equally spaced holes. 4
1, 0, 2, 2, 3, 2, 5, 2, 5, 4, 7, 2, 11, 2, 9, 8, 9, 2, 17, 2, 17, 10, 13, 2, 23, 6, 15, 10, 23, 2, 29, 2, 17, 14, 19, 12, 35, 2, 21, 16, 37, 2, 41, 2, 35, 38, 25, 2, 47, 8, 47, 20, 41, 2, 53, 16, 51, 22, 31, 2, 59, 2, 33, 52, 33, 18, 65, 2, 53, 26, 67, 2, 71, 2, 39, 68, 59, 18, 77, 2, 77, 28, 43, 2, 83, 22, 45, 32, 79 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Holly Krieger and Brady Haran, The Centrifuge Problem, Numberphile video (2018)

T. Y. Lam and K. H. Leung, On vanishing sums for roots of unity, arXiv:math/9511209 [math.NT], 1995.

Matt Parker, The Balanced Centrifuge Problem, Math Blog, 2018.

Gary Sivek, On vanishing sums of distinct roots of unity, #A31, Integers 10 (2010), 365-368.

FORMULA

a(n) = |{ k : k and n-k can be written as a sum of prime factors of n }|.

a(n) = 2 <=> n is prime (A000040).

a(n) >= n-1 <=> n in {1,2,3,4} union { A008588 }.

a(n) = (n+4)/2 <=> n in { A100484 } minus { 4 }.

a(n) = (n+9)/3 <=> n in { A001748 } minus { 9 }.

a(n) = (n+25)/5 <=> n in { A001750 } minus { 25 }.

a(n) = (n+49)/7 <=> n in { A272470 } minus { 49 }.

a(n^2) = n+1 <=> n = 0 or n is prime <=> n in { A182986 }.

a(A001248(n)) = A008864(n).

a(n) is odd <=> n in { A163300 }.

a(n) is even <=> n in { A004280 }.

EXAMPLE

a(6) = |{0,2,3,4,6}| = 5.

a(9) = |{0,3,6,9}| = 4.

a(10) = |{0,2,4,5,6,8,10}| = 7.

MAPLE

a:= proc(n) option remember; local f, b; f, b:=

       map(i-> i[1], ifactors(n)[2]),

       proc(m, i) option remember; m=0 or i>0 and

        (b(m, i-1) or f[i]<=m and b(m-f[i], i))

       end; forget(b); (t-> add(

      `if`(b(j, t) and b(n-j, t), 1, 0), j=0..n))(nops(f))

    end:

seq(a(n), n=0..100);

MATHEMATICA

a[n_] := a[n] = Module[{f, b}, f = FactorInteger[n][[All, 1]]; b[m_, i_] := b[m, i] = m == 0 || i>0 && (b[m, i-1] || f[[i]] <= m && b[m-f[[i]], i]); Function[t, Sum[If[b[j, t] && b[n-j, t], 1, 0], {j, 0, n}]][Length[f]]];

Table[a[n], {n, 0, 100}] (* Jean-Fran├žois Alcover, Dec 13 2018, after Alois P. Heinz *)

CROSSREFS

Cf. A000040, A001248, A001748, A001750, A004280, A008588, A008864, A100484, A103306, A103314, A163300, A182986, A272470, A306275.

Sequence in context: A319810 A325250 A062830 * A164941 A115119 A066656

Adjacent sequences:  A322363 A322364 A322365 * A322367 A322368 A322369

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Dec 04 2018

STATUS

approved

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Last modified April 23 06:08 EDT 2019. Contains 322381 sequences. (Running on oeis4.)