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A283529 The number of partitions of n into simple parts.

%I A283529 #35 Jan 05 2025 19:51:41
%S A283529 1,1,2,2,3,3,5,5,7,7,9,9,12,12,15,15,18,18,22,22,26,26,30,30,35,35,40,
%T A283529 40,45,45,52,52,59,59,66,66,75,75,84,84,93,93,104,104,115,115,126,126,
%U A283529 139,139,152,152,165,165,180,180,195,195,210,210,228,228
%N A283529 The number of partitions of n into simple parts.
%C A283529 Number of partitions of n where each part is simple, meaning that each part is in A002110.
%H A283529 Giovanni Resta, <a href="/A283529/b283529.txt">Table of n, a(n) for n = 0..10000</a>
%H A283529 J. Wang, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/31-4/wang.pdf">Reduced phi-partitions of positive integers</a>, Fib. Quart. 31 (4) (1993) 365-369.
%F A283529 G.f.: 1/Product_{i>=0} (1-x^A002110(i)).
%e A283529 a(6)=5 counts 1+1+1+1+1+1 = 1+1+1+2 = 1+1+2+2 = 2+2+2 =6.
%e A283529 a(7)=5 counts 1+1+1+1+1+1+1 = 1+1+1+1+1+2 = 1+1+1+2+2 = 1+2+2+2 = 1+6.
%p A283529 isA002110 := proc(n)
%p A283529     member(n,[1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 13082761331670030, 614889782588491410, 32589158477190044730, 1922760350154212639070]) ;
%p A283529 end proc:
%p A283529 A283529 := proc(n)
%p A283529     local a,k,issimp,p ;
%p A283529     a := 0 ;
%p A283529     for k in combinat[partition](n) do
%p A283529         issimp := true ;
%p A283529         for p in k do
%p A283529             if not isA002110(p) then
%p A283529                 issimp := false;
%p A283529                 break;
%p A283529             end if;
%p A283529         end do:
%p A283529         if issimp then
%p A283529             a := a+1 ;
%p A283529         end if;
%p A283529     end do:
%p A283529     a ;
%p A283529 end proc:
%t A283529 (* It suffices to compute 3 primorials to get 100 correct terms *)
%t A283529 terms = 100; primorials = FoldList[Times, 1, Prime[Range[3]]]; 1/(Times @@ (1 - x^primorials)) + O[x]^terms // CoefficientList[#, x]& (* _Jean-François Alcover_, May 19 2018 *)
%Y A283529 Cf. A002110, A064986, A283528.
%K A283529 nonn
%O A283529 0,3
%A A283529 _R. J. Mathar_, Mar 10 2017
%E A283529 a(0)=1 prepended by _Alois P. Heinz_, Mar 13 2017