A271384 -id:A271384 - cp's OEIS frontend

A283528 The number of phi-partitions of n.

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

Offset: 1

R. J. Mathar, Mar 10 2017

nonn

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.

			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.

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.

Cf. A000010, A271384, A283530.

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

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 *)
b[n_, m_, i_] := b[n, m, i] = If[n == 0,
     If[m == 0, 1, 0], If[i < 1 || m < 0, 0, b[n, m, i - 1] +
     If[i > n, 0, b[n - i, m - EulerPhi[i], i]]]];
a[n_] := b[n, EulerPhi[n], n]-1;
Array[a, 70] (* Jean-François Alcover, Jun 01 2021, after Alois P. Heinz *)

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

A271382 Least k with precisely n partitions k = x + y satisfying d(k) = d(x) + d(y), where d(k) is the number of divisors of k.

Original entry on oeis.org

2, 14, 10, 26, 44, 45, 126, 68, 99, 104, 162, 117, 98, 124, 232, 164, 148, 200, 260, 333, 231, 244, 248, 297, 273, 284, 315, 406, 332, 345, 385, 430, 344, 399, 388, 436, 429, 488, 465, 495, 472, 525, 561, 555, 621, 556, 632, 604, 652, 712, 536, 693, 735, 675

Offset: 1

Paolo P. Lava, Apr 06 2016

			d(10) = d(1) + d(9) = d(3) + d(7) = d(5) + d(5) = 4 and 10 is the least number with 3 partitions of two numbers with this property: therefore a(3) = 10;
d(126) = d(21) + d(105) = d(22) + d(104) = d(28) + d(98) = d(38) + d(33) = d(40) + d(86) = d(50) + d(76) = d(63) + d(63) = 12 and 126 is the least number with 7 partitions of two numbers with this property: therefore a(7) = 126.

Paolo P. Lava, Table of n, a(n) for n = 1..1200

Cf. A000005, A211224, A271384.

with(numtheory): P:=proc(q) local a,h,k,n; for h from 1 to q do for n from 2*h to q do
a:=0; for k from 1 to trunc(n/2) do if tau(n)=tau(k)+tau(n-k) then a:=a+1; fi; od;
if a=h then print(n); break; fi; od; od; end: P(10^6);

nn = 10^3; Table[SelectFirst[Range@ nn, Function[k, With[{e = DivisorSigma[0, k]}, Count[Transpose@ {Range[k - 1, Ceiling[k/2], -1], Range@ Floor[k/2]}, x_ /; Total@ DivisorSigma[0, x] == e] == n]]], {n, 54}] (* Michael De Vlieger, Apr 06 2016 *)

isok(k, n) = {my(nb = 0, tau = numdiv(k)); for (j=1, k\2, if (numdiv(j)+numdiv(k-j) == tau, nb++); if (nb > n, return (0));); nb == n;}
a(n) = {k=2; while (!isok(k, n), k++); k;} \\ Michel Marcus, Apr 07 2016

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A283528 The number of phi-partitions of n.

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