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

%I A100953 #16 May 21 2021 04:17:22
%S A100953 1,1,0,1,2,5,5,13,14,25,28,54,54,99,105,160,192,295,315,488,546,760,
%T A100953 890,1253,1404,1945,2234,2953,3459,4563,5186,6840,7909,10029,11716,
%U A100953 14843,17123,21635,25035,30981,36098,44581,51370,63259,73223,88739,103048,124752
%N A100953 Number of partitions of n into relatively prime parts such that multiplicities of parts are also relatively prime.
%H A100953 Alois P. Heinz, <a href="/A100953/b100953.txt">Table of n, a(n) for n = 0..10000</a>
%F A100953 Moebius transform of A000837.
%p A100953 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
%p A100953 # second Maple program:
%p A100953 with(numtheory): with(combinat):
%p A100953 b:= proc(n) option remember; `if`(n=0, 1, add(
%p A100953        mobius(n/d)*numbpart(d), d=divisors(n)))
%p A100953     end:
%p A100953 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A100953        mobius(n/d)*b(d), d=divisors(n)))
%p A100953     end:
%p A100953 seq(a(n), n=0..60);  # _Alois P. Heinz_, Dec 19 2017
%t A100953 Table[Length[Select[IntegerPartitions[n],And[GCD@@#===1,GCD@@Length/@Split[#]===1]&]],{n,20}] (* _Gus Wiseman_, Dec 19 2017 *)
%t A100953 b[n_] := b[n] = If[n==0, 1, Sum[
%t A100953      MoebiusMu[n/d]*PartitionsP[d], {d, Divisors[n]}]];
%t A100953 a[n_] := a[n] = If[n==0, 1, Sum[
%t A100953      MoebiusMu[n/d]*b[d], {d, Divisors[n]}]];
%t A100953 a /@ Range[0, 60] (* _Jean-François Alcover_, May 21 2021, after _Alois P. Heinz_ *)
%Y A100953 Cf. A000740, A000837, A007916, A281116, A296302.
%K A100953 easy,nonn
%O A100953 0,5
%A A100953 _Vladeta Jovovic_, Jan 11 2005
%E A100953 More terms from _David Wasserman_ and _R. J. Mathar_, Mar 04 2008
%E A100953 a(0)=1 prepended by _Alois P. Heinz_, Dec 19 2017