[go: up one dir, main page]

login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A257991 Number of odd parts in the partition having Heinz number n. 75
0, 1, 0, 2, 1, 1, 0, 3, 0, 2, 1, 2, 0, 1, 1, 4, 1, 1, 0, 3, 0, 2, 1, 3, 2, 1, 0, 2, 0, 2, 1, 5, 1, 2, 1, 2, 0, 1, 0, 4, 1, 1, 0, 3, 1, 2, 1, 4, 0, 3, 1, 2, 0, 1, 2, 3, 0, 1, 1, 3, 0, 2, 0, 6, 1, 2, 1, 3, 1, 2, 0, 3, 1, 1, 2, 2, 1, 1, 0, 5, 0, 2, 1, 2, 2, 1, 0, 4 (list; graph; refs; listen; history; text; internal format)
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.
In the Maple program the subprogram B yields the partition with Heinz number n.
REFERENCES
George E. Andrews and Kimmo Eriksson, Integer Partitions, Cambridge Univ. Press, Cambridge, 2004.
Miklós Bóna, A Walk Through Combinatorics, World Scientific Publishing Co., 2002.
LINKS
FORMULA
From Amiram Eldar, Jun 17 2024: (Start)
Totally additive with a(p) = 1 if primepi(p) is odd, and 0 otherwise.
a(n) = A257992(n) + A195017(n). (End)
EXAMPLE
a(12) = 2 because the partition having Heinz number 12 = 2*2*3 is [1,1,2], having 2 odd parts.
MAPLE
with(numtheory): a := proc (n) local B, 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: ct := 0: for q to nops(B(n)) do if `mod`(B(n)[q], 2) = 1 then ct := ct+1 else end if end do: ct end proc: seq(a(n), n = 1 .. 135);
# second Maple program:
a:= n-> add(`if`(numtheory[pi](i[1])::odd, i[2], 0), i=ifactors(n)[2]):
seq(a(n), n=1..120); # Alois P. Heinz, May 09 2016
MATHEMATICA
a[n_] := Sum[If[PrimePi[i[[1]]] // OddQ, i[[2]], 0], {i, FactorInteger[n]} ]; Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Dec 10 2016, after Alois P. Heinz *)
CROSSREFS
Sequence in context: A113949 A318808 A349935 * A373592 A343029 A343037
KEYWORD
nonn
AUTHOR
Emeric Deutsch, May 18 2015
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 29 06:09 EDT 2024. Contains 375510 sequences. (Running on oeis4.)