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