A385646 -id:A385646 - cp's OEIS frontend

A385645 a(n) is the number of distinct sums of distinct prime powers dividing n.

1, 3, 3, 7, 3, 6, 3, 15, 7, 7, 3, 10, 3, 7, 7, 31, 3, 13, 3, 12, 7, 7, 3, 18, 7, 7, 15, 14, 3, 11, 3, 63, 7, 7, 7, 19, 3, 7, 7, 20, 3, 13, 3, 15, 14, 7, 3, 34, 7, 15, 7, 15, 3, 27, 7, 22, 7, 7, 3, 15, 3, 7, 14, 127, 7, 13, 3, 15, 7, 13, 3, 27, 3, 7, 15, 15, 7, 13

Offset: 1

Felix Huber, Jul 11 2025

			The a(4) = 7 distinct sums of distinct prime powers dividing 4 are 1, 2, 4, 1 + 2, 1 + 4, 2 + 4 and 1 + 2 + 4.

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

Cf. A000040, A000961, A119347, A385646.

A385645:=proc(n)
    local b,k,l,i,j;
    l:=[1,seq(seq(i[1]^j,j=1..i[2]),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(A385645(n),n=1..78);

a(p) = 3 for prime p.
a(p^k) = A119347(p^k) for prime p and nonnegative integer k.
A385646(n) < a(n) <= A119347(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

A385645 a(n) is the number of distinct sums of distinct prime powers dividing n.

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