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

1,5

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)

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

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 *)

Prog

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

Crossrefs

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

Cf. A079068, A215366.

Sequence in context: A135685 A349447 A164658 * A356676 A160271 A274912

Adjacent sequences: A079064 A079065 A079066 * A079068 A079069 A079070

Keyword

nonn,easy

Author

Reinhard Zumkeller, Dec 20 2002

Status

approved