A023888 Sum of prime power divisors of n (1 included).

1, 3, 4, 7, 6, 6, 8, 15, 13, 8, 12, 10, 14, 10, 9, 31, 18, 15, 20, 12, 11, 14, 24, 18, 31, 16, 40, 14, 30, 11, 32, 63, 15, 20, 13, 19, 38, 22, 17, 20, 42, 13, 44, 18, 18, 26, 48, 34, 57, 33, 21, 20, 54, 42, 17, 22, 23, 32, 60, 15, 62, 34, 20, 127, 19, 17, 68, 24, 27

Offset: 1

Author

Olivier Gérard

Keywords

• nonn

Comments

Sum of n-th row of triangle A210208. [Reinhard Zumkeller, Mar 18 2012]

Examples

			For n = 12, set of such divisors is {1, 2, 3, 4}; a(12) = 1+2+3+4 = 10. From
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A008475, A159077.

Programs

Haskell

a023888 = sum . a210208_row  -- Reinhard Zumkeller, Mar 18 2012

Maple

f:= n -> 1 + add((t[1]^(t[2]+1)-t[1])/(t[1]-1),t=ifactors(n)[2]):
map(f, [$1..100]); # Robert Israel, Jan 04 2017

Mathematica

Array[ Plus @@ (Select[ Divisors[ # ], (Length[ FactorInteger[ # ] ]<=1)& ])&, 70 ]

PARI

for(n=1,100, s=1; fordiv(n,d, if((ispower(d,,&z)&&isprime(z)) || isprime(d),s+=d)); print1(s,", "))

PARI

a(n) = {
  my(f = factor(n), fsz = matsize(f)[1]);
  1 + sum(k = 1, fsz, f[k,1]*(f[k,1]^f[k,2] - 1)\(f[k,1]-1));
};
vector(100, n, a(n))  \\ Gheorghe Coserea, Jan 04 2017

Formula

a(n) = A000203(n) - A035321(n) = A023889(n) + 1.

a(1) = 1, a(p) = p+1, a(pq) = p+q+1, a(pq...z) = (p+q+...+z) + 1, a(p^k) = (p^(k+1)-1) / (p-1), for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.

G.f.: x/(1 - x) + Sum_{k>=2} floor(1/omega(k))*k*x^k/(1 - x^k), where omega(k) is the number of distinct prime factors (A001221). - Ilya Gutkovskiy, Jan 04 2017