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
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..10000
Programs
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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 *)
Formula
Moebius transform of A000837.
Extensions
More terms from David Wasserman and R. J. Mathar, Mar 04 2008
a(0)=1 prepended by Alois P. Heinz, Dec 19 2017