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

1,2

Links

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

Formula

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

Example

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.

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 *)

Prog

(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])); }

Crossrefs

Cf. A000203, A002191, A027750, A240698.

Sequence in context: A355593 A355302 A084302 * A301855 A080256 A351394

Adjacent sequences: A289869 A289870 A289871 * A289873 A289874 A289875

Keyword

nonn

Author

Michel Marcus, Jul 14 2017

Status

approved