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 A023888 Sum of prime power divisors of n (1 included). 10

%I

%S 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,

%T 40,14,30,11,32,63,15,20,13,19,38,22,17,20,42,13,44,18,18,26,48,34,57,

%U 33,21,20,54,42,17,22,23,32,60,15,62,34,20,127,19,17,68,24,27

%N Sum of prime power divisors of n (1 included).

%C Sum of n-th row of triangle A210208. [_Reinhard Zumkeller_, Mar 18 2012]

%H Reinhard Zumkeller, <a href="/A023888/b023888.txt">Table of n, a(n) for n = 1..10000</a>

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

%F 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.

%F 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

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

%p f:= n -> 1 + add((t[1]^(t[2]+1)-t[1])/(t[1]-1),t=ifactors(n)[2]):

%p map(f, [\$1..100]); # _Robert Israel_, Jan 04 2017

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

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

%o a023888 = sum . a210208_row -- _Reinhard Zumkeller_, Mar 18 2012

%o (PARI)

%o a(n) = {

%o my(f = factor(n), fsz = matsize(f)[1]);

%o 1 + sum(k = 1, fsz, f[k,1]*(f[k,1]^f[k,2] - 1)\(f[k,1]-1));

%o };

%o vector(100, n, a(n)) \\ _Gheorghe Coserea_, Jan 04 2017

%Y Cf. A008475, A159077.

%K nonn

%O 1,2