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A283528 The number of phi-partitions of n. 6
0, 0, 1, 1, 2, 0, 3, 4, 8, 2, 4, 1, 5, 8, 24, 24, 6, 2, 7, 15, 107, 46, 8, 4, 135, 101, 347, 83, 9, 0, 10, 460, 1019, 431, 1308, 13, 11, 842, 2760, 214, 12, 2, 13, 1418, 5124, 2977, 14, 42, 2021, 720, 17355, 4997, 15, 70, 21108, 3674, 40702, 16907, 16, 1, 17 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

The number of partitions n = a1+a2+...+ak which have at least two parts and obey phi(n) = phi(a1)+phi(a2)+...+phi(ak). phi(.) = A000010(.) is Euler's totient. The trivial result with one part, n=a1, is not counted; that would induce another sequence with terms a(n)+1.

LINKS

Giovanni Resta and Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 100 terms from Giovanni Resta)

P. Jones, Phi-partitions, Fib. Quart. 29 (4) (1991) 347-350.

C. Powell, On the uniqueness of reduced phi-partitions, Fib. Quart. 34 (3) (1996) 194-199.

J. Wang, Reduced phi-partitions of positive integers, Fib. Quart. 31 (4) (1993) 365-369.

EXAMPLE

a(7) = 3 counts the partitions 1+1+1+1+1+2 = 1+1+1+1+3 = 1+1+5.

a(8) = 4 counts the partitions 2+2+2+2 = 2+2+4 = 4+4 = 1+1+6.

MAPLE

A283528 := proc(n)

    local a, k, phip ;

    a := 0 ;

    for k in combinat[partition](n) do

        if 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:

# second Maple program:

with(numtheory):

b:= proc(n, m, i) option remember; `if`(n=0,

      `if`(m=0, 1, 0), `if`(i<1 or m<0, 0, b(n, m, i-1)+

      `if`(i>n, 0, b(n-i, m-phi(i), i))))

    end:

a:= n-> b(n, phi(n), n)-1:

seq(a(n), n=1..70);  # Alois P. Heinz, Mar 10 2017

MATHEMATICA

Table[ Length@ IntegerPartitions[n 10^7 + EulerPhi[n], {2, Infinity},

EulerPhi@ Range[n-1] + 10^7 Range[n-1]], {n, 60}] (* Giovanni Resta, Mar 10 2017 *)

CROSSREFS

Cf. A000010, A271384, A283530.

Sequence in context: A243202 A257136 A258871 * A110990 A254213 A321171

Adjacent sequences:  A283525 A283526 A283527 * A283529 A283530 A283531

KEYWORD

nonn

AUTHOR

R. J. Mathar, Mar 10 2017

EXTENSIONS

a(56)-a(61) from Giovanni Resta, Mar 10 2017

STATUS

approved

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Last modified January 27 21:50 EST 2020. Contains 331300 sequences. (Running on oeis4.)