A385645 -id:A385645 - cp's OEIS frontend

A385646 a(n) is the number of distinct sums of distinct prime factors of n.

0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 3, 1, 1, 3, 1, 3, 3, 3, 1, 3, 1, 3, 1, 3, 1, 6, 1, 1, 3, 3, 3, 3, 1, 3, 3, 3, 1, 7, 1, 3, 3, 3, 1, 3, 1, 3, 3, 3, 1, 3, 3, 3, 3, 3, 1, 6, 1, 3, 3, 1, 3, 7, 1, 3, 3, 6, 1, 3, 1, 3, 3, 3, 3, 7, 1, 3, 1, 3, 1, 7, 3, 3, 3, 3

Offset: 1

Felix Huber, Jul 11 2025

			The a(18) = 3 distinct sums of distinct prime factors of 18 = 2*3^2 are 2, 3 and 2 + 3.
The a(42) = 7 distinct sums of distinct prime factors of 42 = 2*3*7 are 2, 3, 7, 2 + 3 = 5, 2 + 7 = 9, 3 + 7 = 10, 2 + 3 + 7 = 12.
The a(30) = 6 distinct sums of distinct prime factors of 30 = 2*3*5 are 2, 3, 2 + 3 = 5, 2 + 5 = 7, 3 + 5 = 8, 2 + 3 + 5 = 10.

Felix Huber, Table of n, a(n) for n = 1..10000

Cf. A000040, A001221, A119347, A385645.

A385646:=proc(n)
    local b,k,l,i,j;
    l:=[seq(i[1],i in ifactors(n)[2])]:
    b:=proc(m,i)
        option remember;
        `if`(m=0,1,`if`(i<1,0,b(m,i-1)+`if`(l[i]>m,0,b(m-l[i],i-1))))
    end;
    return nops(select(x->x>0,[seq(b(k,nops(l)),k=1..add(l))]))
    end:
seq(A385646(n),n=1..85);

a(n) < A385646(n).

A385904 a(n) is the number of nonempty subsets of the divisors of n that sum to a perfect square.

Original entry on oeis.org

1, 1, 2, 2, 1, 4, 1, 3, 3, 2, 1, 11, 1, 3, 4, 5, 1, 9, 1, 9, 3, 3, 1, 27, 2, 2, 4, 8, 1, 27, 1, 7, 3, 2, 2, 49, 1, 1, 3, 22, 1, 21, 1, 7, 8, 3, 1, 77, 2, 5, 2, 4, 1, 22, 2, 21, 2, 1, 1, 248, 1, 2, 7, 11, 1, 21, 1, 4, 2, 17, 1, 235, 1, 1, 9, 7, 1, 20, 1, 64, 6, 1

Offset: 1

Felix Huber, Jul 21 2025

nonn

			a(6) = 4 because exactly the 4 nonempty subsets {1}, {1, 3}, {1, 2, 6} and {3, 6} of the divisors of 6 sum to a perfect square: 1 = 1^2, 1 + 3 = 2^2, 1 + 2 + 6 = 3^2.

Felix Huber, Table of n, a(n) for n = 1..10000

Cf. A000290, A005835, A027750, A119347, A119348, A237290, A385645, A385646.

with(NumberTheory):
A385904:=proc(n)
    local b,l,j;
    l:=[(Divisors(n))[]]:
    b:=proc(m,i)
        option remember;
        `if`(m=0,1,`if`(i<1,0,b(m,i-1)+`if`(l[i]>m,0,b(m-l[i],i-1))))
	end;
    add(b(j^2,nops(l)),j=1..floor(sqrt(sigma(n))));
end:
seq(A385904(n),n=1..82);

a[n_]:=Module[{nb = 0, d = Divisors[n]},Length[Select[Subsets[d],IntegerQ[Sqrt[Total[#]]]&]]]-1;Array[a,82] (* James C. McMahon, Jul 27 2025 *)

a(n) = my(nb=0, d=divisors(n)); forsubset(#d, s, nb+=issquare(sum(i=1, #s, d[s[i]]))); nb-1; \\ Michel Marcus, Jul 22 2025

a(p) = 1 for primes p != 3.

cp's OEIS Frontend

A385646 a(n) is the number of distinct sums of distinct prime factors of n.

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