A023888 -id:A023888 - 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

A073093 Number of prime power divisors of n.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Aug 24 2002

Also, number of prime divisors of 2n (counted with multiplicity).
A001221(n) < a(n) <= A000005(n) for all n; a(n)=A001221(n)+1 iff n is squarefree (A005117); a(n)=A000005(n) iff n is a prime power (A000961).
a(n) is also the number of kBenoit Cloitre, Oct 13 2002
a(n) is also 1 + the number of divisors of n with omega(d)=1, where omega is A001221. - Enrique Pérez Herrero, Nov 05 2009
Length of n-th row of triangle A210208. - Reinhard Zumkeller, Mar 18 2012
a(n) depends only on the prime signature of n with a(A025487(n)) = 1, 2, 3, 3, 4, 4, 5, 5, 4, 6, 5, 6, 5, 7, 6, 7 ,.. = A036041(n)+1; (n>=1). - R. J. Mathar, May 28 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
T. M. Apostol, Resultants of Cyclotomic Polynomials, Proc. Amer. Math. Soc. 24, 457-462, 1970.
T. M. Apostol, The Resultant of the Cyclotomic Polynomials Fm(ax) and Fn(bx), Math. Comput. 29, 1-6, 1975.
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

Cf. A000961, A023888, A054372. Bisection of A001222.

a073093 = length . a210208_row  -- Reinhard Zumkeller, Mar 18 2012

[n eq 1 select 1 else &+[p[2]: p in Factorization(n)]+1: n in [1..100]]; // Vincenzo Librandi, Jan 06 2017

seq(numtheory:-bigomega(n)+1, n=1..1000); # Robert Israel, Sep 06 2015

f[n_] := Plus @@ Flatten[ Table[1, {#[[2]]}] & /@ FactorInteger[n]]; Table[ f[2n], {n, 105}] (* Robert G. Wilson v, Dec 23 2004 *)
A001221[n_] := (Length[ FactorInteger[n]]); SetAttributes[A001221, Listable]; A073093[n_]:=Length[Select[A001221[Divisors[n]], # == 1 &]]; (* Enrique Pérez Herrero, Nov 05 2009 *)
PrimeOmega[Range[100]] + 1 (* Paolo Xausa, Nov 23 2024 *)

numlib::Omega (2*n)$ n=1..105 // Zerinvary Lajos, May 13 2008

a(n)=sum(k=1,n,if(1-polresultant(polcyclo(n),polcyclo(k)),1,0))

A073093(n)=bigomega(n)+1   \\ M. F. Hasler, Dec 08 2010

If n = Product (p_j^k_j), a(n) = 1 + Sum (k_j).
a(n) = bigomega(n)+1 = A001222(n)+1 = A001222(2*n).
a(n) = if n=1 then 1 else a(A032742(n)) + 1. - Reinhard Zumkeller, Sep 24 2009
a(n) = max { a(d) ; d 1. - David W. Wilson, Dec 08 2010
a(n) = Sum_{k = 1 .. A001221(n)} A010055(A027750(n,k)). - Reinhard Zumkeller, Mar 18 2012
G.f.: x/(1 - x) + Sum_{k>=2} floor(1/omega(k))*x^k/(1 - x^k), where omega(k) is the number of distinct prime factors (A001221). - Ilya Gutkovskiy, Jan 04 2017

A023889 Sum of the prime power divisors of n (not including 1).

Original entry on oeis.org

0, 2, 3, 6, 5, 5, 7, 14, 12, 7, 11, 9, 13, 9, 8, 30, 17, 14, 19, 11, 10, 13, 23, 17, 30, 15, 39, 13, 29, 10, 31, 62, 14, 19, 12, 18, 37, 21, 16, 19, 41, 12, 43, 17, 17, 25, 47, 33, 56, 32, 20, 19, 53, 41, 16, 21, 22, 31, 59, 14, 61, 33, 19, 126, 18, 16, 67, 23, 26, 14

Offset: 1

Olivier Gérard

nonn

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A001221, A023888.

Array[ Plus @@ (Select[ Divisors[ # ], PrimePowerQ ])&, 80 ]

a(n) = sumdiv(n, d, if(isprimepower(d), d)); \\ Michel Marcus, Mar 21 2017; corrected by Daniel Suteu, Jul 20 2018

a(n) = my(f = factor(n)); sum(k = 1, #f~, f[k, 1] * (f[k, 1]^f[k, 2] - 1) / (f[k, 1] - 1)) \\ Daniel Suteu, Jul 20 2018

G.f.: 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
a(n) = A023888(n) - 1. - Michel Marcus, Mar 21 2017
a(n) = Sum_{d|n} d * [omega(d) = 1], where omega is the number of distinct prime factors and [ ] is the Iverson bracket. - Wesley Ivan Hurt, Jan 28 2021

A210208 Triangle read by rows in which row n lists the divisors of n that are prime powers, A000961.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 3, 1, 7, 1, 2, 4, 8, 1, 3, 9, 1, 2, 5, 1, 11, 1, 2, 3, 4, 1, 13, 1, 2, 7, 1, 3, 5, 1, 2, 4, 8, 16, 1, 17, 1, 2, 3, 9, 1, 19, 1, 2, 4, 5, 1, 3, 7, 1, 2, 11, 1, 23, 1, 2, 3, 4, 8, 1, 5, 25, 1, 2, 13, 1, 3, 9, 27, 1, 2, 4, 7

Offset: 1

Reinhard Zumkeller, Mar 18 2012

{T(n,k): k = 1..A073093(n)} subset of {A027750(n,k): k = 1..A000005(n)} for all n.

			Table begins:
  1;
  1, 2;
  1, 3;
  1, 2, 4;
  1, 5;
  1, 2, 3;
  1, 7;
  1, 2, 4, 8;
  1, 3, 9;
  1, 2, 5;
  1, 11;
  1, 2, 3, 4; - _Geoffrey Critzer_, Feb 08 2015

Reinhard Zumkeller, Rows n=1..5000 of triangle, flattened
Eric Weisstein's World of Mathematics, Divisor.

Cf. A073093 (row lengths), A023888 (row sums), A034699 (row maxima), A183091 (row products).

Cf. A000005, A010055, A027750, A141809.

a210208 n k = a210208_tabf !! (n-1) !! (n-1)
a210208_row n = a210208_tabf !! (n-1)
a210208_tabf = map (filter ((== 1) . a010055)) a027750_tabf

Table[Prepend[Select[Divisors[n], PrimeNu[#] == 1 &], 1], {n, 1, 10}]//Grid (* Geoffrey Critzer, Feb 08 2015 *)

row(n) = select(x -> omega(x) < 2, divisors(n)); \\ Amiram Eldar, May 02 2025

A034699(n) = T(n,A073093(n)) = maximum of n-th row.

A183091 a(n) is the product of the divisors p^k of n where p is prime and k >= 1.

Original entry on oeis.org

1, 2, 3, 8, 5, 6, 7, 64, 27, 10, 11, 24, 13, 14, 15, 1024, 17, 54, 19, 40, 21, 22, 23, 192, 125, 26, 729, 56, 29, 30, 31, 32768, 33, 34, 35, 216, 37, 38, 39, 320, 41, 42, 43, 88, 135, 46, 47, 3072, 343, 250, 51, 104, 53, 1458

Offset: 1

Jaroslav Krizek, Dec 25 2010

Product 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 = 24.

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

Cf. A007955, A023888, A153038, A183092, A210208.

a183091 = product . a210208_row  -- Reinhard Zumkeller, Mar 18 2012

A183091 := proc(n) local a,d; a := 1 ;for d in numtheory[divisors](n) minus {1} do  if nops( numtheory[factorset](d)) = 1 then a := a*d; end if; end do: a ; end proc: # R. J. Mathar, Apr 14 2011

Table[Product[d, {d, Select[Divisors[n], Length[FactorInteger[#]] == 1 &]}], {n,1, 54}] (* Geoffrey Critzer, Mar 18 2015 *)
f[p_, e_] := p^(e*(e+1)/2); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 31 2023 *)

a(n)=my(f=factor(n)); prod(i=1,#f~, f[i,1]^binomial(f[i,2]+1,2)) \\ Charles R Greathouse IV, Nov 11 2014

a(n) = A007955(n) / A183092(n).
Multiplicative with a(p^k) = p^(k*(k+1)/2).
The Dirichlet g.f. of a(n) / abs(A153038(n)) is Product_{k >= 0} zeta(s+k). - Álvar Ibeas, Nov 10 2014

A159077 a(n) = A008475(n) + 1.

Original entry on oeis.org

1, 3, 4, 5, 6, 6, 8, 9, 10, 8, 12, 8, 14, 10, 9, 17, 18, 12, 20, 10, 11, 14, 24, 12, 26, 16, 28, 12, 30, 11, 32, 33, 15, 20, 13, 14, 38, 22, 17, 14, 42, 13, 44, 16, 15, 26, 48, 20, 50, 28, 21, 18, 54, 30, 17, 16, 23, 32, 60, 13, 62, 34, 17, 65, 19, 17, 68, 22, 27, 15, 72, 18, 74

Offset: 1

Jaroslav Krizek, Apr 04 2009

nonn

If n = Product (p_i^k_i) for i = 1, …, j then a(n) is sum of divisor d from set of divisors{1, p_1^k_1, p_2^k_2, …, p_j^k_j}.

			For n = 12, set of divisors {1, p_1^k_1, p_2^k_2, …, p_j^k_j}: {1, 3, 4}. a(12) = 1+3+4=8.

Cf. A008475, A023888.

f[n_] := 1 + Plus @@ Power @@@ FactorInteger@ n; f[1] = 1; Array[f, 60]

a(n)=local(t); if(n<1, 0, t=factor(n); 1+sum(k=1, matsize(t)[1], t[k, 1]^t[k, 2])) /* Anton Mosunov, Jan 05 2017 */

a(n) = [Sum_(i=1,…, j) p_i^k_i] + 1 = A000203(n) - A178636(n).

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, for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.

Edited by N. J. A. Sloane, Apr 07 2009

A284117 Sum of proper prime power divisors of n.

Original entry on oeis.org

0, 0, 0, 4, 0, 0, 0, 12, 9, 0, 0, 4, 0, 0, 0, 28, 0, 9, 0, 4, 0, 0, 0, 12, 25, 0, 36, 4, 0, 0, 0, 60, 0, 0, 0, 13, 0, 0, 0, 12, 0, 0, 0, 4, 9, 0, 0, 28, 49, 25, 0, 4, 0, 36, 0, 12, 0, 0, 0, 4, 0, 0, 9, 124, 0, 0, 0, 4, 0, 0, 0, 21, 0, 0, 25, 4, 0, 0, 0, 28, 117, 0, 0, 4, 0, 0, 0, 12, 0, 9, 0, 4, 0, 0, 0, 60, 0, 49, 9, 29

Offset: 1

Ilya Gutkovskiy, Mar 20 2017

			a(8) = 12 because 12 has 6 divisors {1, 2, 3, 4, 6, 12} among which 2 are proper prime powers {4, 8} therefore 4 + 8 = 12.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of divisors.

Cf. A001414, A008472, A023888, A023889, A046660, A246547.

f:= n -> add(t[1]*(t[1]^t[2]-t[1])/(t[1]-1), t=ifactors(n)[2]):
map(f, [$1..100]); # Robert Israel, Mar 31 2017

nmax = 100; Rest[CoefficientList[Series[Sum[Boole[PrimePowerQ[k] && PrimeOmega[k] > 1] k x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Total[Select[Divisors[n], PrimePowerQ[#1] && PrimeOmega[#1] > 1 &]], {n, 100}]
f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - p - 1; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 24 2024 *)

concat([0, 0, 0], Vec(sum(k=1, 100, (isprimepower(k) && bigomega(k)>1) * k * x^k/(1 - x^k)) + O(x^101))) \\ Indranil Ghosh, Mar 21 2017

a(n) = sumdiv(n, d, d*(isprimepower(d) && !isprime(d))); \\ Michel Marcus, Apr 01 2017

G.f.: Sum_{p prime, k>=2} p^k*x^(p^k)/(1 - x^(p^k)).
a(n) = Sum_{d|n, d = p^k, p prime, k >= 2} d.
a(n) = 0 if n is a squarefree (A005117).
Additive with a(p^e) = (p^(e+1)-1)/(p-1) - p - 1. - Amiram Eldar, Jul 24 2024

A286875 If n = Product (p_j^k_j) then a(n) = Sum (k_j >= 2, p_j^k_j).

Original entry on oeis.org

0, 0, 0, 4, 0, 0, 0, 8, 9, 0, 0, 4, 0, 0, 0, 16, 0, 9, 0, 4, 0, 0, 0, 8, 25, 0, 27, 4, 0, 0, 0, 32, 0, 0, 0, 13, 0, 0, 0, 8, 0, 0, 0, 4, 9, 0, 0, 16, 49, 25, 0, 4, 0, 27, 0, 8, 0, 0, 0, 4, 0, 0, 9, 64, 0, 0, 0, 4, 0, 0, 0, 17, 0, 0, 25, 4, 0, 0, 0, 16, 81, 0, 0, 4, 0, 0, 0, 8, 0, 9, 0, 4, 0, 0, 0, 32, 0, 49, 9, 29, 0, 0, 0, 8, 0, 0, 0, 31

Offset: 1

Ilya Gutkovskiy, Aug 02 2017

Sum of unitary, proper prime power divisors of n.

			a(360) = a(2^3*3^2*5) = 2^3 + 3^2 = 17.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sums of divisors.

Cf. A005117, A008475, A222416, A023888, A023889, A034448, A063956, A077610, A092261, A246547, A284117.

Table[DivisorSum[n, # &, CoprimeQ[#, n/#] && PrimePowerQ[#] && !PrimeQ[#] &], {n, 108}]
f[p_, e_] := If[e == 1, 0, p^e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 24 2024 *)

A286875(n) = { my(f=factor(n)); for (i=1, #f~, if(f[i, 2] < 2, f[i, 1] = 0)); vecsum(vector(#f~,i,f[i,1]^f[i,2])); }; \\ Antti Karttunen, Oct 07 2017

from sympy import primefactors, isprime, gcd, divisors
def a(n): return sum(d for d in divisors(n) if gcd(d, n//d)==1 and len(primefactors(d))==1 and not isprime(d))
print([a(n) for n in range(1, 109)]) # Indranil Ghosh, Aug 02 2017

a(n) = Sum_{d|n, d = p^k, p prime, k >= 2, gcd(d, n/d) = 1} d.
a(A246547(k)) = A246547(k).
a(A005117(k)) = 0.
Additive with a(p^e) = p^e if e >= 2, and 0 otherwise. - Amiram Eldar, Jul 24 2024

A367502 Sum of the final digits of the prime power divisors (p^k, k>=0) of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 6, 8, 15, 13, 8, 2, 10, 4, 10, 9, 21, 8, 15, 10, 12, 11, 4, 4, 18, 11, 6, 20, 14, 10, 11, 2, 23, 5, 10, 13, 19, 8, 12, 7, 20, 2, 13, 4, 8, 18, 6, 8, 24, 17, 13, 11, 10, 4, 22, 7, 22, 13, 12, 10, 15, 2, 4, 20, 27, 9, 7, 8, 14, 7, 15, 2, 27, 4, 10, 14, 16, 9, 9, 10, 26, 21, 4, 4

Offset: 1

Wesley Ivan Hurt, Nov 20 2023

			a(16) = 21 since the prime power divisors of 16 are {1, 2, 4, 8, 16} and the sum of their final digits is 1 + 2 + 4 + 8 + 6 = 21.

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

Cf. A000961 (Powers of primes), A001221 (omega), A010879 (Final digit of n), A023888 (Sum of the prime power divisors of n including 1), A371885 (first k with a(k) = n).

f:= proc(n) local F,i,j,t;
  F:= ifactors(n)[2];
  1 + add(add(F[i,1]^j mod 10, j = 1 .. F[i,2]),i=1..nops(F))
end proc:
map(f, [$1..100]); # Robert Israel, Apr 10 2024

Table[1 + Sum[Floor[1/PrimeNu[k]] Mod[k, 10] (1 - Ceiling[n/k] + Floor[n/k]), {k, 2, n}], {n, 100}]

a(n) = my(f=factor(n)); 1 + sum(k=1, #f~, sum(j=1, f[k,2], lift(Mod(f[k,1], 10)^j))); \\ Michel Marcus, Nov 22 2023

a(n) = 1 + Sum_{d|n, d>1} floor(1/omega(d)) * (d mod 10).

A284233 Sum of odd prime power divisors of n (not including 1).

Original entry on oeis.org

0, 0, 3, 0, 5, 3, 7, 0, 12, 5, 11, 3, 13, 7, 8, 0, 17, 12, 19, 5, 10, 11, 23, 3, 30, 13, 39, 7, 29, 8, 31, 0, 14, 17, 12, 12, 37, 19, 16, 5, 41, 10, 43, 11, 17, 23, 47, 3, 56, 30, 20, 13, 53, 39, 16, 7, 22, 29, 59, 8, 61, 31, 19, 0, 18, 14, 67, 17, 26, 12, 71, 12, 73, 37, 33, 19, 18, 16, 79, 5

Offset: 1

Ilya Gutkovskiy, Mar 23 2017

			a(15) = 8 because 15 has 4 divisors {1, 3, 5, 15} among which 2 are odd prime powers {3, 5} therefore 3 + 5 = 8.

Eric Weisstein's World of Mathematics, Prime Power.
Index entries for sequences related to sums of divisors.

Cf. A000961, A005069, A023888, A023889, A038712, A061345, A065091 (fixed points), A087436 (number of odd prime power divisors of n), A206787, A246655, A284117.

nmax = 80; Rest[CoefficientList[Series[Sum[Boole[PrimePowerQ[k] && Mod[k, 2] == 1] k x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Total[Select[Divisors[n], PrimePowerQ[#] && Mod[#, 2] == 1 &]], {n, 80}]
f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; f[2, e_] := 0; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 24 2024 *)

G.f.: Sum_{k>=1} A061345(k)*x^A061345(k)/(1 - x^A061345(k)).
a(n) = Sum_{d|n, d = p^k, p prime, p > 2, k > 0} d.
a(p^k) = p*(p^k - 1)/(p - 1) for p is a prime > 2.
a(2^k*p) = p for p is a prime > 2.
a(2^k) = 0.
Additive with a(2^e) = 0, and a(p^e) = (p^(e+1)-1)/(p-1) - 1 for an odd prime p. - Amiram Eldar, Jul 24 2024

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).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

PARI

Python

Formula

A073093 Number of prime power divisors of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

MuPAD

PARI

PARI

Formula

A023889 Sum of the prime power divisors of n (not including 1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A210208 Triangle read by rows in which row n lists the divisors of n that are prime powers, A000961.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A183091 a(n) is the product of the divisors p^k of n where p is prime and k >= 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

A159077 a(n) = A008475(n) + 1.

Views

Author

Keywords

Comments

Examples

Crossrefs