A283530 The number of reduced phi-partitions of n.

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, 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, 12, 8, 5, 1, 5, 1, 14, 13, 14, 5, 3, 1, 13, 9, 16, 1, 1

Offset: 1

Author

R. J. Mathar, Mar 10 2017

Keywords

• nonn

Comments

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.

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.

Examples

			a(15)=2 counts 1+2+2+2+2+2+2= 1+1+1+2+2+2+6.
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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Giovanni Resta)
• J. Wang, Reduced phi-partitions of positive integers, Fib. Quart. 31 (4) (1993) 365-369.
• J. Wang, X. Wang, On the set of reduced phi-partitions of a positive integer, Fib. Quart. 44 (2) (2006) 98-102.

Crossrefs

Cf. A283528, A283529, A283320.

Programs

Maple

isA002110 := proc(n)
    member(n,[1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 13082761331670030, 614889782588491410, 32589158477190044730, 1922760350154212639070]) ;
end proc:
A283530 := proc(n)
    local a,k,issimp,p ;
    a := 0 ;
    for k in combinat[partition](n) do
        issimp := true ;
        for p in k do
            if not isA002110(p) then
                issimp := false;
                break;
            end if;
        end do:
        if issimp and nops(k) > 1 then
            phip := add(numtheory[phi](p),p=k) ;
            if phip = numtheory[phi](n) then
                a := a+1 ;
            end if;
        end if;
    end do:
    a ;
end proc:

Mathematica

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 *)

Formula

a(A002110(k)) = 0. [Wang]