A289872 a(n) is the number of partial sums of the divisors of n that are the sum of divisors of some integer.

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 4, 2, 3, 3, 5, 2, 5, 2, 5, 3, 4, 2, 6, 3, 3, 4, 6, 2, 5, 2, 6, 4, 4, 4, 4, 2, 3, 3, 7, 2, 6, 2, 6, 4, 3, 2, 6, 3, 5, 3, 5, 2, 6, 3, 6, 3, 4, 2, 8, 2, 3, 4, 7, 3, 6, 2, 5, 3, 7, 2, 6, 2, 4, 4, 4, 3, 6, 2, 7, 5, 4, 2, 6, 3, 3, 3, 6, 2, 6

Offset: 1

Author

Michel Marcus, Jul 14 2017

Keywords

• nonn

Examples

			For n=2, the divisors are 1, 2; the partial sums are 1, 3; 1=sigma(1) and 3=sigma(2); so a(2)=2.
For n=10, the divisors are 1, 2, 5, 10; the partial sums are 1, 3, 8, 18; 1=sigma(1), 3=sigma(2), 8=sigma(7) and 18=sigma(10); so a(10)=4.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203, A002191, A027750, A240698.

Programs

Maple

M:= 1000: # get a(n) for n=1..m where m is the first number with sigma(m+1) > M
S:= Vector(M):
for n from 1 to M-1 do
  v:= numtheory:-sigma(n);
  if v > M then if not assigned(nmax) then nmax:= n-1 fi
  elif S[v] = 0 then S[v]:= 1
  fi;
od:
seq(add(S[i],i=ListTools:-PartialSums(sort(convert(numtheory:-divisors(n),list)))), n = 1..nmax); # Robert Israel, Jul 14 2017

Mathematica

s = Union@ DivisorSigma[1, Range[10^6]]; Array[Count[Accumulate@ Divisors@ #, k_ /; MemberQ[s, k]] &, 90] (* Michael De Vlieger, Jul 14 2017 *)

PARI

issigma(n) = {for (k=1, n, if (sigma(k) == n, return (1));); 0;}
a(n) = {d = divisors(n); v = vector(#d, k, sum(j=1, k, d[j])); sum(k=1, #v, issigma(v[k]));}

Formula

For n>=1 and p prime, a(p^n) = n+1.