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A283530 The number of reduced phi-partitions of n.

%I A283530 #28 Jan 05 2025 19:51:41
%S A283530 0,0,1,1,1,0,1,2,1,1,1,1,1,2,2,3,1,1,1,2,3,3,1,1,2,3,3,3,1,0,1,5,4,4,
%T A283530 3,2,1,5,4,4,1,1,1,6,6,7,1,3,2,6,5,8,1,3,4,8,6,10,1,1,1,11,9,12,5,2,1,
%U A283530 12,8,5,1,5,1,14,13,14,5,3,1,13,9,16,1,1
%N A283530 The number of reduced phi-partitions of n.
%C A283530 The reduced phi-partitions of n are partitions n= a_1 +a_2 +a_3 +... +a_k into at least 2 parts such that each part is simple (i.e. each part in A002110, as required in A283529) and such that in addition phi(n) = sum_i phi(a_i), as required in A283528. phi(.) = A000010(.) is Euler's totient.
%C A283530 Numbers n where a(n)=1 are called semisimple. 3, 4, 5, 7, 9, 10, 11, 12, 13, 17, 18, 19, 23, 24,... are semisimple (see A283320). In this list of semisimple numbers there are no odd numbers besides 9 and the odd primes.
%H A283530 Alois P. Heinz, <a href="/A283530/b283530.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Giovanni Resta)
%H A283530 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.
%H A283530 J. Wang, X. Wang, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/44-2/quartwang02_2006.pdf">On the set of reduced phi-partitions of a positive integer</a>, Fib. Quart. 44 (2) (2006) 98-102.
%F A283530 a(A002110(k)) = 0. [Wang]
%e A283530 a(15)=2 counts 1+2+2+2+2+2+2= 1+1+1+2+2+2+6.
%e A283530 a(16)=3 counts 2+2+2+2+2+2+2+2 = 1+1+2+2+2+2+6 = 1+1+1+1+6+6.
%p A283530 isA002110 := proc(n)
%p A283530     member(n,[1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 13082761331670030, 614889782588491410, 32589158477190044730, 1922760350154212639070]) ;
%p A283530 end proc:
%p A283530 A283530 := proc(n)
%p A283530     local a,k,issimp,p ;
%p A283530     a := 0 ;
%p A283530     for k in combinat[partition](n) do
%p A283530         issimp := true ;
%p A283530         for p in k do
%p A283530             if not isA002110(p) then
%p A283530                 issimp := false;
%p A283530                 break;
%p A283530             end if;
%p A283530         end do:
%p A283530         if issimp and nops(k) > 1 then
%p A283530             phip := add(numtheory[phi](p),p=k) ;
%p A283530             if phip = numtheory[phi](n) then
%p A283530                 a := a+1 ;
%p A283530             end if;
%p A283530         end if;
%p A283530     end do:
%p A283530     a ;
%p A283530 end proc:
%t A283530 v={1,2,6,30,210}; e=10^9 v + EulerPhi@v; a[n_] := Length@ IntegerPartitions[ 10^9 n + EulerPhi[n], {2, Infinity}, e]; Array[a, 100] (* suitable for n <= 1000, _Giovanni Resta_, Mar 10 2017 *)
%Y A283530 Cf. A283528, A283529, A283320.
%K A283530 nonn
%O A283530 1,8
%A A283530 _R. J. Mathar_, Mar 10 2017