A258120 Number of Fibonacci numbers in the partition having Heinz number n.

0, 1, 1, 2, 1, 2, 0, 3, 2, 2, 1, 3, 0, 1, 2, 4, 0, 3, 1, 3, 1, 2, 0, 4, 2, 1, 3, 2, 0, 3, 0, 5, 2, 1, 1, 4, 0, 2, 1, 4, 1, 2, 0, 3, 3, 1, 0, 5, 0, 3, 1, 2, 0, 4, 2, 3, 2, 1, 0, 4, 0, 1, 2, 6, 1, 3, 0, 2, 1, 2, 0, 5, 1, 1, 3, 3, 1, 2, 0, 5, 4

Offset

1,4

Comments

We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436; consequently, a(2436) = 3.

The subprogram B of the Maple program gives the partition having Heinz number n.

a(m*n) = a(m)+a(n).

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Example

a(2)=1 because B(2)=[1]; a(3)=1 because B(3)=[2]; a(4)=2 because B(4)=[1,1]; a(28)=2 because B(28)=[1,1,4].

Maple

with(numtheory): a := proc (n) local B, F, ct, q: B := proc (n) local nn, j, m; nn := op(2, ifactors(n)): for j to nops(nn) do m[j] := op(j, nn) end do: [seq(seq(pi(op(1, m[i])), q = 1 .. op(2, m[i])), i = 1 .. nops(nn))] end proc; F := {seq(combinat['fibonacci'](1+i), i = 1 .. max(B(n)))}: ct := 0; for q to nops(B(n)) do if member(B(n)[q], F) = true then ct := ct+1 else  end if end do: ct end proc: seq(a(n), n = 1 .. 150);

Mathematica

B[n_] := Module[{nn, j, m}, nn = FactorInteger[n]; For[j = 1, j <= Length[nn], j++, m[j] = nn[[j]]]; Flatten[ Table[ Table[ PrimePi[ m[i][[1]]], {q, 1, m[i][[2]]}], {i, 1, Length[nn]}]]];
a[n_] := Module[{F, ct, q}, F = Union @ Table[Fibonacci[1 + i], {i, 1, Max[ B[n]]}]; ct = 0; For[q = 1, q <= Length[B[n]], q++, If[MemberQ[F, B[n][[q]]], ct++]]; ct];
Table[a[n], {n, 1, 150}] (* Jean-François Alcover, Apr 25 2017, translated from Maple *)

Crossrefs

Cf. A000045, A215366.

Sequence in context: A332104 A238735 A356006 * A379311 A379306 A147786

Adjacent sequences: A258117 A258118 A258119 * A258121 A258122 A258123

Keyword

nonn

Author

Emeric Deutsch, Jun 14 2015

Status

approved