A100953 Number of partitions of n into relatively prime parts such that multiplicities of parts are also relatively prime.

1, 1, 0, 1, 2, 5, 5, 13, 14, 25, 28, 54, 54, 99, 105, 160, 192, 295, 315, 488, 546, 760, 890, 1253, 1404, 1945, 2234, 2953, 3459, 4563, 5186, 6840, 7909, 10029, 11716, 14843, 17123, 21635, 25035, 30981, 36098, 44581, 51370, 63259, 73223, 88739, 103048, 124752

Offset

0,5

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Formula

Moebius transform of A000837.

Maple

read transforms : a000837 := [] : b000837 := fopen("b000837.txt", READ) : bfil := readline(b000837) : while StringTools[WordCount](bfil) > 0 do b := sscanf( bfil, "%d %d") ; a000837 := [op(a000837), op(2, b)] ; bfil := readline(b000837) ; od: fclose(b000837) ; a000837 := subsop(1=NULL, a000837) : a := MOBIUS(a000837) : for n from 1 to 120 do printf("%d, ", op(n, a)) ; od: # R. J. Mathar, Mar 12 2008
# second Maple program:
with(numtheory): with(combinat):
b:= proc(n) option remember; `if`(n=0, 1, add(
       mobius(n/d)*numbpart(d), d=divisors(n)))
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(
       mobius(n/d)*b(d), d=divisors(n)))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 19 2017

Mathematica

Table[Length[Select[IntegerPartitions[n], And[GCD@@#===1, GCD@@Length/@Split[#]===1]&]], {n, 20}] (* Gus Wiseman, Dec 19 2017 *)
b[n_] := b[n] = If[n==0, 1, Sum[
     MoebiusMu[n/d]*PartitionsP[d], {d, Divisors[n]}]];
a[n_] := a[n] = If[n==0, 1, Sum[
     MoebiusMu[n/d]*b[d], {d, Divisors[n]}]];
a /@ Range[0, 60] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

Crossrefs

Cf. A000740, A000837, A007916, A281116, A296302.

Sequence in context: A183719 A239340 A124201 * A112835 A206625 A176168

Adjacent sequences: A100950 A100951 A100952 * A100954 A100955 A100956

Keyword

easy,nonn

Author

Vladeta Jovovic, Jan 11 2005

Extensions

More terms from David Wasserman and R. J. Mathar, Mar 04 2008

a(0)=1 prepended by Alois P. Heinz, Dec 19 2017

Status

approved