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A258120
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Number of Fibonacci numbers in the partition having Heinz number n.
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1
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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
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OFFSET
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1,4
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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].
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MAPLE
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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);
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MATHEMATICA
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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];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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