login
A079067
Number of primes less than greatest prime factor of n but not dividing n.
12
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.
Sequence in context: A135685 A349447 A164658 * A356676 A160271 A274912
KEYWORD
nonn,easy
AUTHOR
Reinhard Zumkeller, Dec 20 2002
STATUS
approved