A159077 -id:A159077 - cp's OEIS frontend

A008475 If n = Product (p_j^k_j) then a(n) = Sum (p_j^k_j) (a(1) = 0 by convention).

0, 2, 3, 4, 5, 5, 7, 8, 9, 7, 11, 7, 13, 9, 8, 16, 17, 11, 19, 9, 10, 13, 23, 11, 25, 15, 27, 11, 29, 10, 31, 32, 14, 19, 12, 13, 37, 21, 16, 13, 41, 12, 43, 15, 14, 25, 47, 19, 49, 27, 20, 17, 53, 29, 16, 15, 22, 31, 59, 12, 61, 33, 16, 64, 18, 16, 67, 21, 26, 14, 71, 17, 73

Offset: 1

Olivier Gérard

For n>1, a(n) is the minimal number m such that the symmetric group S_m has an element of order n. - Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 26 2001

If gcd(u,w) = 1, then a(u*w) = a(u) + a(w); behaves like logarithm; compare A001414 or A056239. - Labos Elemer, Mar 31 2003

			a(180) = a(2^2 * 3^2 * 5) = 2^2 + 3^2 + 5 = 18.

F. J. Budden, The Fascination of Groups, Cambridge, 1972; pp. 322, 573.
József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter IV, p. 147.
T. Z. Xuan, On some sums of large additive number theoretic functions (in Chinese), Journal of Beijing normal university, No. 2 (1984), pp. 11-18.

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from T. D. Noe)
John Bamberg, Grant Cairns and Devin Kilminster, The crystallographic restriction, permutations and Goldbach's conjecture, Amer. Math. Monthly, Vol. 110, No. 3 (March 2003), pp. 202-209.
Roger B. Eggleton and William P. Galvin, Upper Bounds on the Sum of Principal Divisors of an Integer, Mathematics Magazine, Vol. 77, No. 3 (Jun., 2004), pp. 190-200.

Cf. A001414, A000961, A005117, A051613, A072691, A081402, A081403, A081404, A027748, A124010, A001221, A028233, A034684, A053585, A159077, A023888, A078771, A092509, A286875.

See A222416 for the variant with a(1)=1.

a008475 1 = 0
a008475 n = sum $ a141809_row n
-- Reinhard Zumkeller, Jan 29 2013, Oct 10 2011

A008475 := proc(n) local e,j; e := ifactors(n)[2]:
add(e[j][1]^e[j][2], j=1..nops(e)) end:
seq(A008475(n), n=1..60); # Peter Luschny, Jan 17 2010

f[n_] := Plus @@ Power @@@ FactorInteger@ n; f[1] = 0; Array[f, 73]

for(n=1,100,print1(sum(i=1,omega(n), component(component(factor(n),1),i)^component(component(factor(n),2),i)),","))

a(n)=local(t);if(n<1,0,t=factor(n);sum(k=1,matsize(t)[1],t[k,1]^t[k,2])) /* Michael Somos, Oct 20 2004 */

A008475(n) = { my(f=factor(n)); vecsum(vector(#f~,i,f[i,1]^f[i,2])); }; \\ Antti Karttunen, Nov 17 2017

from sympy import factorint
def a(n):
    f=factorint(n)
    return 0 if n==1 else sum([i**f[i] for i in f]) # Indranil Ghosh, May 20 2017

Additive with a(p^e) = p^e.
a(A000961(n)) = A000961(n); a(A005117(n)) = A001414(A005117(n)).
a(n) = Sum_{k=1..A001221(n)} A027748(n,k) ^ A124010(n,k) for n>1. - Reinhard Zumkeller, Oct 10 2011
a(n) = Sum_{k=1..A001221(n)} A141809(n,k) for n > 1. - Reinhard Zumkeller, Jan 29 2013
Sum_{k=1..n} a(k) ~ (Pi^2/12)* n^2/log(n) + O(n^2/log(n)^2) (Xuan, 1984). - Amiram Eldar, Mar 04 2021

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

Original entry on oeis.org

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

Olivier Gérard

nonn

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

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

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

Cf. A008475, A159077.

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

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

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

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

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

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

A178636 If n = Product (p_i^k_i) for i = 1, ..., j then a(n) is the sum of the divisors d that are not in the set {1, p_1^k_1, p_2^k_2, ..., p_j^k_j}.

Original entry on oeis.org

0, 0, 0, 2, 0, 6, 0, 6, 3, 10, 0, 20, 0, 14, 15, 14, 0, 27, 0, 32, 21, 22, 0, 48, 5, 26, 12, 44, 0, 61, 0, 30, 33, 34, 35, 77, 0, 38, 39, 76, 0, 83, 0, 68, 63, 46, 0, 104, 7, 65, 51, 80, 0, 90, 55, 104, 57, 58, 0, 155, 0, 62, 87, 62, 65, 127, 0, 104, 69, 129, 0, 177, 0, 74, 95, 116, 77, 149, 0, 164, 39, 82, 0, 209, 85, 86, 87, 160, 0, 217, 91, 140, 93, 94, 95, 216, 0, 119, 135, 187

Offset: 1

Jaroslav Krizek, Dec 25 2010

nonn

			For n = 12, set of divisors {1, p_1^k_1, p_2^k_2, ..., p_j^k_j}: {1, 3, 4}. Complement of divisors: {2, 6, 12}. a(12) = 2+6+12 = 20.

a(n) = A000203(n) - A159077(n) = A167515(n) - 1.

a(1) = 0, a(p) = 0, a(pq) = pq, a(pq...z) = [(p+1)* (q+1)* ... *(z+1)] - [p+q+ ...+z] - 1, a(p^k) = (p^k-p)/(p-1), for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.

I edited the definition to fix the grammar and make it understandable.

a(100) corrected by Georg Fischer, Dec 10 2022

cp's OEIS Frontend

A008475 If n = Product (p_j^k_j) then a(n) = Sum (p_j^k_j) (a(1) = 0 by convention).

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

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A178636 If n = Product (p_i^k_i) for i = 1, ..., j then a(n) is the sum of the divisors d that are not in the set {1, p_1^k_1, p_2^k_2, ..., p_j^k_j}.

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