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.

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

Offset

0,3

Comments

Numbers where a(n) + A000010(n) != n + 1: A102467. - Robert G. Wilson v, Aug 23 2021

Links

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

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

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.

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

Robert G. Wilson v, Graph of the first 100001 terms.

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

$RecursionLimit = 4096;
a[1] = 0;
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]);
     With[{t = Length[f]}, Sum[
     If[b[j, t] && b[n - j, t], 1, 0], {j, 0, n}]]];
Table[a[n], {n, 0, 1000}] (* Jean-François Alcover, Dec 13 2018, after Alois P. Heinz, corrected and updated Aug 07 2021 *)
f[n_] := Block[{c = 2, k = 2, p = First@# & /@ FactorInteger@ n}, While[k < n, If[ IntegerPartitions[k, All, p, 1] != {} && IntegerPartitions[n - k, All, p, 1] != {}, c++]; k++]; c]; f[0] = 1; f[1] = 0; Array[f, 75] (* Robert G. Wilson v, Aug 22 2021 *)

Crossrefs

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

Sequence in context: A319810 A325250 A062830 * A363724 A345268 A164941

Adjacent sequences: A322363 A322364 A322365 * A322367 A322368 A322369

Keyword

nonn,look

Author

Alois P. Heinz, Dec 04 2018

Status

approved