A079067 Number of primes less than greatest prime factor of n but not dividing n.

0, 0, 1, 0, 2, 0, 3, 0, 1, 1, 4, 0, 5, 2, 1, 0, 6, 0, 7, 1, 2, 3, 8, 0, 2, 4, 1, 2, 9, 0, 10, 0, 3, 5, 2, 0, 11, 6, 4, 1, 12, 1, 13, 3, 1, 7, 14, 0, 3, 1, 5, 4, 15, 0, 3, 2, 6, 8, 16, 0, 17, 9, 2, 0, 4, 2, 18, 5, 7, 1, 19, 0, 20, 10, 1, 6, 3, 3, 21, 1, 1, 11, 22, 1, 5, 12, 8, 3, 23, 0, 4, 7, 9, 13, 6, 0

Offset: 1

Author

Reinhard Zumkeller, Dec 20 2002

Keywords

• nonn
• easy

Comments

For n >= 2, a(n) is the largest part minus the number of distinct parts of the partition having Heinz number n. The Heinz number of a partition [i_1, i_2, ..., i_r] is defined as Product_{j=1..r} (i_j-th prime) (concept used by Alois P. Heinz in A215366 as an encoding of a partition). For example, for the partition [1, 1, 1, 4] we get 2*2*2*7 = 56; a(56) = 4 - #{1,4} = 2. - Emeric Deutsch, Jun 09 2015 [edited by Peter Munn, Apr 09 2024]

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from G. C. Greubel)

Crossrefs

See the formula section for the relationships with A000040, A001221, A002110, A006530, A049084, A061395, A083720.

Cf. A079068, A215366.

Programs

Maple

with(numtheory): a := proc (n) local B: 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: max(B(n))-nops(convert(B(n), set)) end proc: 0, seq(a(n), n = 2 .. 96); # The subprogram B yields the partition having Heinz number n. # Emeric Deutsch, Jun 09 2015
# second Maple program:
with(numtheory):
a:= n-> (s-> pi(max(0, s))-nops(s))(factorset(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Sep 03 2019

Mathematica

a[1] = 0; a[n_] := With[{fi = FactorInteger[n]}, PrimePi[fi][[-1, 1]] - Length[fi]]; Array[a, 100] (* Jean-François Alcover, Jan 08 2016 *)

PARI

a(n) = if (n==1, 0, my(pf=factor(n)[,1]); primepi(vecmax(pf)) - #pf); \\ Michel Marcus, May 05 2017

Formula

a(n) = A049084(A006530(n)) - A001221(n) = A061395(n) - A001221(n).

a(n) = 0 iff n = m*prime(k)#, where prime(k)# is the k-th primorial (A002110(k)) and A006530(m) <= A000040(k).

a(A000040(k)) = k - 1.

a(n) = A001221(A083720(n)). - Peter Munn, Apr 09 2024