A351247 -id:A351247 - cp's OEIS frontend

A001221 Number of distinct primes dividing n (also called omega(n)).

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 3, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 3, 2, 1, 2, 1, 3, 2

Offset: 1

N. J. A. Sloane

From Peter C. Heinig (algorithms(AT)gmx.de), Mar 08 2008: (Start)
This is also the number of maximal ideals of the ring (Z/nZ,+,*). Since every finite integral domain must be a field, every prime ideal of Z/nZ is a maximal ideal and since in general each maximal ideal is prime, there are just as many prime ideals as maximal ones in Z/nZ, so the sequence gives the number of prime ideals of Z/nZ as well.
The reason why this number is given by the sequence is that the ideals of Z/nZ are precisely the subgroups of (Z/nZ,+). Hence for an ideal to be maximal it has form a maximal subgroup of (Z/nZ,+) and this is equivalent to having prime index in (Z/nZ) and this is equivalent to being generated by a single prime divisor of n.
Finally, all the groups arising in this way have different orders, hence are different, so the number of maximal ideals equals the number of distinct primes dividing n. (End)
Equals double inverse Mobius transform of A143519, where A051731 = the inverse Mobius transform. - Gary W. Adamson, Aug 22 2008
a(n) is the number of unitary prime power divisors of n (not including 1). - Jaroslav Krizek, May 04 2009 [corrected by Ilya Gutkovskiy, Oct 09 2019]
Sum_{d|n} 2^(-A001221(d) - A001222(n/d)) = Sum_{d|n} 2^(-A001222(d) - A001221(n/d)) = 1 (see Dressler and van de Lune link). - Michel Marcus, Dec 18 2012
Up to 2*3*5*7*11*13*17*19*23*29  - 1 = 6469693230 - 1, also the decimal expansion of the constant 0.01111211... = Sum_{k>=0} 1/(10 ^ A000040(k) - 1) (see A073668). - Eric Desbiaux, Jan 20 2014
The average order of a(n): Sum_{k=1..n} a(k) ~ Sum_{k=1..n} log log k. - Daniel Forgues, Aug 13-16 2015
From Peter Luschny, Jul 13 2023: (Start)
We can use A001221 and A001222 to classify the positive integers as follows.
   A001222(n) = A001221(n) = 0 singles out {1}.
Restricting to n > 1:
   A001222(n)^A001221(n) = 1: A000040, prime numbers.
   A001221(n)^A001222(n) = 1: A246655, prime powers.
   A001222(n)^A001221(n) > 1: A002808, the composite numbers.
   A001221(n)^A001222(n) > 1: A024619, complement of A246655.
   n^(A001222(n) - A001221(n)) = 1: A144338, products of distinct primes. (End)
Inverse Möbius transform of the characteristic function of primes (A010051). - Wesley Ivan Hurt, Jun 22 2024
Dirichlet convolution of A010051(n) and 1. - Wesley Ivan Hurt, Jul 15 2025

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, pp. 48-57.
J. Peters, A. Lodge and E. J. Ternouth, E. Gifford, Factor Table (n<100000) (British Association Mathematical Tables Vol.V), Burlington House/Cambridge University Press London 1935.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane and Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 from NJAS)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Henry Bottomley, Prime factors calculator
J. Brennen, Prime Factoring Applet
J. Britton, Prime Factorization Machine
A. Dendane, Prime Factors Calculator
Robert E. Dressler and Jan van de Lune, Some remarks concerning the number theoretic functions omega and Omega, Proc. Amer. Math. Soc. 41 (1973), 403-406
J. Flament, Decomposition d'un nombre en facteurs premiers
G. H. Hardy and S. Ramanujan, The normal number of prime factors of a number, Quart. J. Math. 48 (1917), 76-92. Also Collected papers of Srinivasa Ramanujan, AMS Chelsea Publ., Providence, RI (2000): 262-275.
A. Hodges, Java Applet for Factorisation
M. Kac, Statistical Independence in Probability, Analysis and Number Theory, Carus Monograph 12, Math. Assoc. Amer., 1959, see p. 64.
S. O. S. Math, Prime factorization of the first 1000 integers
K. Matthews, Factorization and calculating phi(n),omega(n),d(n),sigma(n) and mu(n)
J. Moyer, "Prime Factors of Integers" server for numbers up to 10^36
Primefan, The First 2500 Integers,Factored
Primefan, Factorer
F. Richman, Factoring into Primes
Eric Weisstein's World of Mathematics, Distinct Prime Factors
Eric Weisstein's World of Mathematics, Moebius Transform
Eric Weisstein's World of Mathematics, Prime Factor
Eric Weisstein's World of Mathematics, Prime zeta function primezeta(s).
Wikipedia, Prime factor
Wikipedia, Table of prime factors
G. Xiao, WIMS server, Factoris
Index entries for "core" sequences
Index entries for sequences computed from exponents in factorization of n

Cf. A001222 (primes counted with multiplicity), A046660, A285577, A346617. Partial sums give A013939.
Cf. also A125666, A069010, A087624, A091202, A091221, A143519, A144494, A158210, A156542, A156552, A000010, A008683.
Sum of the k-th powers of the primes dividing n for k=0..10: this sequence (k=0), A008472 (k=1), A005063 (k=2), A005064 (k=3), A005065 (k=4), A351193 (k=5), A351194 (k=6), A351195 (k=7), A351196 (k=8), A351197 (k=9), A351198 (k=10).
Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k=0..10: this sequence (k=0), A069359 (k=1), A322078 (k=2), A351242 (k=3), A351244 (k=4), A351245 (k=5), A351246 (k=6), A351247 (k=7), A351248 (k=8), A351249 (k=9), A351262 (k=10).

import Math.NumberTheory.Primes.Factorisation (factorise)
a001221 = length . snd . unzip . factorise
-- Reinhard Zumkeller, Nov 28 2015

using Nemo
function NumberOfPrimeFactors(n; distinct=true)
    distinct && return length(factor(ZZ(n)))
    sum(e for (p, e) in factor(ZZ(n)); init=0)
end
println([NumberOfPrimeFactors(n) for n in 1:60]) # Peter Luschny, Jan 02 2024

[#PrimeDivisors(n): n in [1..120]]; // Bruno Berselli, Oct 15 2021

A001221 := proc(n) local t1, i; if n = 1 then return 0 else t1 := 0; for i to n do if n mod ithprime(i) = 0 then t1 := t1 + 1 end if end do end if; t1 end proc;
A001221 := proc(n) nops(numtheory[factorset](n)) end proc: # Emeric Deutsch
omega := n -> NumberTheory:-NumberOfPrimeFactors(n, 'distinct'): # Peter Luschny, Jun 15 2025

Array[ Length[ FactorInteger[ # ] ]&, 100 ]
PrimeNu[Range[120]]  (* Harvey P. Dale, Apr 26 2011 *)

func(nops(numlib::primedivisors(n)), n):

numlib::omega(n)$ n=1..110 // Zerinvary Lajos, May 13 2008

PARI
```
a(n)=omega(n)
```

from sympy.ntheory import primefactors
print([len(primefactors(n)) for n in range(1, 1001)])  # Indranil Ghosh, Mar 19 2017

def A001221(n): return sum(1 for p in divisors(n) if is_prime(p))
[A001221(n) for n in (1..80)] # Peter Luschny, Feb 01 2012

[sloane.A001221(n) for n in (1..111)] # Giuseppe Coppoletta, Jan 19 2015

[gp.omega(n) for n in range(1,101)] # G. C. Greubel, Jul 13 2024

G.f.: Sum_{k>=1} x^prime(k)/(1-x^prime(k)). - Benoit Cloitre, Apr 21 2003; corrected by Franklin T. Adams-Watters, Sep 01 2009
Dirichlet generating function: zeta(s)*primezeta(s). - Franklin T. Adams-Watters, Sep 11 2005
Additive with a(p^e) = 1.
a(1) = 0, a(p) = 1, a(pq) = 2, a(pq...z) = k, a(p^k) = 1, where p, q, ..., z are k distinct primes and k natural numbers. - Jaroslav Krizek, May 04 2009
a(n) = log_2(Sum_{d|n} mu(d)^2). - Enrique Pérez Herrero, Jul 09 2012
a(A002110(n)) = n, i.e., a(prime(n)#) = n. - Jean-Marc Rebert, Jul 23 2015
a(n) = A091221(A091202(n)) = A069010(A156552(n)). - Antti Karttunen, circa 2004 & Mar 06 2017
L.g.f.: -log(Product_{k>=1} (1 - x^prime(k))^(1/prime(k))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jul 30 2018
a(n) = log_2(Sum_{k=1..n} mu(gcd(n,k))^2/phi(n/gcd(n,k))) = log_2(Sum_{k=1..n} mu(n/gcd(n,k))^2/phi(n/gcd(n,k))), where phi = A000010 and mu = A008683. - Richard L. Ollerton, May 13 2021
Sum_{k=1..n} 2^(-a(gcd(n,k)) - A001222(n/gcd(n,k)))/phi(n/gcd(n,k)) = Sum_{k=1..n} 2^(-A001222(gcd(n,k)) - a(n/gcd(n,k)))/phi(n/gcd(n,k)) = 1, where phi = A000010. - Richard L. Ollerton, May 13 2021
a(n) = A005089(n) + A005091(n) + A059841(n) = A005088(n) +A005090(n) +A079978(n). - R. J. Mathar, Jul 22 2021
From Wesley Ivan Hurt, Jun 22 2024: (Start)
a(n) = Sum_{p|n, p prime} 1.
a(n) = Sum_{d|n} c(d), where c = A010051. (End)

A069359 a(n) = n * Sum_{p|n} 1/p where p are primes dividing n.

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 4, 3, 7, 1, 10, 1, 9, 8, 8, 1, 15, 1, 14, 10, 13, 1, 20, 5, 15, 9, 18, 1, 31, 1, 16, 14, 19, 12, 30, 1, 21, 16, 28, 1, 41, 1, 26, 24, 25, 1, 40, 7, 35, 20, 30, 1, 45, 16, 36, 22, 31, 1, 62, 1, 33, 30, 32, 18, 61, 1, 38, 26, 59, 1, 60, 1, 39, 40, 42, 18, 71, 1, 56

Offset: 1

Benoit Cloitre, Apr 15 2002

nonn

Coincides with arithmetic derivative on squarefree numbers: a(A005117(n)) = A068328(n) = A003415(A005117(n)). - Reinhard Zumkeller, Jul 20 2003, Clarified by Antti Karttunen, Nov 15 2019
a(n) = n-1 iff n = 1 or n is a primary pseudoperfect number A054377. - Jonathan Sondow, Apr 16 2014
a(1) = 0 by the standard convention for empty sums.
“Seva” on the MathOverflow link asks if the iterates of this sequence are all eventually 0. - Charles R Greathouse IV, Feb 15 2019

			a(12) = 10 because the prime divisors of 12 are 2 and 3 so we have: 12/2 + 12/3 = 6 + 4 = 10. - _Geoffrey Critzer_, Mar 17 2015

Antti Karttunen, Table of n, a(n) for n = 1..16384 (first 10000 terms from Franklin T. Adams-Watters)
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
MathOverflow, A recursion with a number-theoretic function (2019)
Joshua Zelinsky, The sum of the reciprocals of the prime divisors of an odd perfect or odd primitive non-deficient number, arXiv:2402.14234 [math.NT], 2024. See also Integers (2025) Vol. 25, Art. No. A59, page 2.

Cf. A003415, A005117, A068328, A010051, A000027, A054377, A180253, A230593, A292786, A306369, A326690, A329029, A329350, A329352.
Cf. A322068 (partial sums), A323599 (Inverse Möbius transform).
Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k = 0..10: A001221 (k=0), this sequence (k=1), A322078 (k=2), A351242 (k=3), A351244 (k=4), A351245 (k=5), A351246 (k=6), A351247 (k=7), A351248 (k=8), A351249 (k=9), A351262 (k=10).

[0] cat [n*&+[1/p: p in PrimeDivisors(n)]:n in [2..80]]; // Marius A. Burtea, Jan 21 2020

A069359 := n -> add(n/d, d = select(isprime, numtheory[divisors](n))):
seq(A069359(i), i = 1..20); # Peter Luschny, Jan 31 2012
# second Maple program:
a:= n-> n*add(1/i[1], i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 23 2019

f[list_, i_] := list[[i]]; nn = 100; a = Table[n, {n, 1, nn}]; b =
Table[If[PrimeQ[n], 1, 0], {n, 1, nn}]; Table[DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}] (* Geoffrey Critzer, Mar 17 2015 *)

a(n) = n*sumdiv(n, d, isprime(d)/d); \\ Michel Marcus, Mar 18 2015

a(n) = my(ps=factor(n)[,1]~);sum(k=1,#ps,n\ps[k]) \\ Franklin T. Adams-Watters, Apr 09 2015

from sympy import primefactors
def A069359(n): return sum(n//p for p in primefactors(n)) # Chai Wah Wu, Feb 05 2022

def A069359(n) :
    D = filter(is_prime, divisors(n))
    return add(n/d for d in D)
print([A069359(i) for i in (1..20)]) # Peter Luschny, Jan 31 2012

G.f.: Sum(x^p(j)/(1-x^p(j))^2,j>=1), where p(j) is the j-th prime. - Vladeta Jovovic, Mar 29 2006
a(n) = A230593(n) - n. a(n) = A010051(n) (*) A000027(n), where operation (*) denotes Dirichlet convolution, that is, convolution of type: a(n) = Sum_(d|n) b(d) * c(n/d) = Sum_{d|n} A010051(d) * A000027(n/d). - Jaroslav Krizek, Nov 07 2013
a(A054377(n)) = A054377(n) - 1. - Jonathan Sondow, Apr 16 2014
Dirichlet g.f.: zeta(s - 1)*primezeta(s). - Geoffrey Critzer, Mar 17 2015
Sum_{k=1..n} a(k) ~ A085548 * n^2 / 2. - Vaclav Kotesovec, Feb 04 2019
From Antti Karttunen, Nov 15 2019: (Start)
a(n) = Sum_{d|n} A008683(n/d)*A323599(d).
a(n) = A003415(n) - A329039(n) = A230593(n) - n = A306369(n) - A000010(n).
a(n) = A276085(A329350(n)) = A048675(A329352(n)).
a(A276086(n)) = A329029(n), a(A328571(n)) = A329031(n).
(End)
a(n) = Sum_{d|n} A000010(d) * A001221(n/d). - Torlach Rush, Jan 21 2020
a(n) = Sum_{k=1..n} omega(gcd(n, k)). - Ilya Gutkovskiy, Feb 21 2020
a(p^k) = p^(k-1) for p prime and k>=1. - Wesley Ivan Hurt, Jul 15 2025

A322078 a(n) = n^2 * Sum_{p|n, p prime} 1/p^2.

Original entry on oeis.org

0, 1, 1, 4, 1, 13, 1, 16, 9, 29, 1, 52, 1, 53, 34, 64, 1, 117, 1, 116, 58, 125, 1, 208, 25, 173, 81, 212, 1, 361, 1, 256, 130, 293, 74, 468, 1, 365, 178, 464, 1, 673, 1, 500, 306, 533, 1, 832, 49, 725, 298, 692, 1, 1053, 146, 848, 370, 845, 1, 1444, 1, 965, 522

Offset: 1

Daniel Suteu, Nov 25 2018

nonn

Generalized formula is f(n,m) = n^m * Sum_{p|n} 1/p^m, where f(n,0) = A001221(n) and f(n,1) = A069359(n).

Dirichlet convolution of A010051(n) and n^2. - Wesley Ivan Hurt, Jul 15 2025

			a(40) = 464 because the prime factors of 40 are 2 and 5, so we have 40^2 * (1/2^2 + 1/5^2) = 464.

Daniel Suteu, Table of n, a(n) for n = 1..10000

Cf. A000040, A000330, A001221, A007434, A010051, A069359, A085541.

Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k = 0..10: A001221 (k=0), A069359 (k=1), this sequence (k=2), A351242 (k=3), A351244 (k=4), A351245 (k=5), A351246 (k=6), A351247 (k=7), A351248 (k=8), A351249 (k=9), A351262 (k=10).

[0] cat [n^2*&+[1/p^2:p in PrimeDivisors(n)]:n in [2..70]]; // Marius A. Burtea, Oct 10 2019

a:= n-> n^2*add(1/i[1]^2, i=ifactors(n)[2]):
seq(a(n), n=1..70);  # Alois P. Heinz, Oct 11 2019

f[p_, e_] := 1/p^2; a[n_] := If[n==1, 0, n^2*Plus@@f@@@FactorInteger[n]]; Array[a, 60] (* Amiram Eldar, Nov 26 2018 *)

a(n) = my(f=factor(n)[,1]~); sum(k=1, #f, n^2\f[k]^2);

Sum_{k=1..n} a(k) ~ A085541 * A000330(n).
G.f.: Sum_{k>=1} x^prime(k) * (1 + x^prime(k)) / (1 - x^prime(k))^3. - Ilya Gutkovskiy, Oct 10 2019
a(n) = 1 <=> n is prime.  _Alois P. Heinz, Oct 11 2019
Dirichlet g.f.: zeta(s-2)*primezeta(s).  This follows because Sum_{n>=1} a(n)/n^s = Sum_{n>=1} (n^2/n^s) Sum_{p|n} 1/p^2. Since n = p*j, rewrite the sum as Sum_{p} Sum_{j>=1} 1/(p^2*(p*j)^(s-2)) = Sum_{p} 1/p^s Sum_{j>=1} 1/j^(s-2) = zeta(s-2)*primezeta(s). The result generalizes to higher powers of p. - Michael Shamos, Mar 02 2023
a(n) = Sum_{d|n} A007434(d)*A001221(n/d). - Ridouane Oudra, Jul 13 2025
From Wesley Ivan Hurt, Jul 15 2025: (Start)
a(n) = Sum_{d|n} c(d) * (n/d)^2, where c = A010051.
a(p^k) = p^(2*k-2) for p prime and k>=1. (End)

A351242 a(n) = n^3 * Sum_{p|n, p prime} 1/p^3.

Original entry on oeis.org

0, 1, 1, 8, 1, 35, 1, 64, 27, 133, 1, 280, 1, 351, 152, 512, 1, 945, 1, 1064, 370, 1339, 1, 2240, 125, 2205, 729, 2808, 1, 4591, 1, 4096, 1358, 4921, 468, 7560, 1, 6867, 2224, 8512, 1, 12221, 1, 10712, 4104, 12175, 1, 17920, 343, 16625, 4940, 17640, 1, 25515, 1456, 22464

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

Dirichlet convolution of A010051(n) and n^3. - Wesley Ivan Hurt, Jul 15 2025

			a(6) = 35; a(6) = 6^3 * Sum_{p|6, p prime} 1/p^3 = 216 * (1/2^3 + 1/3^3) = 35.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k = 0..10: A001221 (k=0), A069359 (k=1), A322078 (k=2), this sequence (k=3), A351244 (k=4), A351245 (k=5), A351246 (k=6), A351247 (k=7), A351248 (k=8), A351249 (k=9), A351262 (k=10).

Cf. A000040, A001221, A010051, A059376, A085964.

a[n_]:= n^3 * Sum[1/p^3,{p,Select[Divisors[n],PrimeQ]}]; Array[a,56] (* Stefano Spezia, Jul 14 2025 *)

a(n) = my(f = factor(n)); sum(i = 1, #f~, (n/f[i,1])^3) \\ David A. Corneth, Jul 14 2025

a(A000040(n)) = 1.
G.f.: Sum_{k>=1} x^prime(k) * (1 + 4*x^prime(k) + x^(2*prime(k))) / (1 - x^prime(k))^4. - Ilya Gutkovskiy, Feb 05 2022
Dirichlet g.f. = zeta(s-3)*primezeta(s).  This follows because Sum_{n>=1} a(n)/n^s = Sum_{n>=1} (n^3/n^s) Sum_{p|n} 1/p^3. Since n = p*j, rewrite the sum as Sum_{p} Sum_{j>=1} 1/(p^3*(p*j)^(s-3)) = Sum_{p} 1/p^s Sum_{j>=1} 1/j^(s-3) = zeta(s-3)*primezeta(s).  The result generalizes to higher powers of p in a(n). - Michael Shamos, Mar 01 2023
Sum_{k=1..n} a(k) ~ A085964 * n^4/4. - Vaclav Kotesovec, Mar 03 2023
a(n) = Sum_{d|n} A059376(d)*A001221(n/d). - Ridouane Oudra, Jul 14 2025
From Wesley Ivan Hurt, Jul 15 2025: (Start)
a(n) = Sum_{d|n} c(d) * (n/d)^3, where c = A010051.
a(p^k) = p^(3*k-3) for p prime and k>=1. (End)

A351244 a(n) = n^4 * Sum_{p|n, p prime} 1/p^4.

Original entry on oeis.org

0, 1, 1, 16, 1, 97, 1, 256, 81, 641, 1, 1552, 1, 2417, 706, 4096, 1, 7857, 1, 10256, 2482, 14657, 1, 24832, 625, 28577, 6561, 38672, 1, 61921, 1, 65536, 14722, 83537, 3026, 125712, 1, 130337, 28642, 164096, 1, 234193, 1, 234512, 57186, 279857, 1, 397312, 2401, 400625, 83602

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

nonn

Dirichlet convolution of A010051(n) and n^4. - Wesley Ivan Hurt, Jul 15 2025

			a(6) = 97; a(6) = 6^4 * Sum_{p|6, p prime} 1/p^4 = 1296 * (1/2^4 + 1/3^4) = 97.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k = 0..10: A001221 (k=0), A069359 (k=1), A322078 (k=2), A351242 (k=3), this sequence (k=4), A351245 (k=5), A351246 (k=6), A351247 (k=7), A351248 (k=8), A351249 (k=9), A351262 (k=10).

Cf. A000040, A001221, A010051, A059377, A085965.

a[n_]:= n^4 * Sum[1/p^4, {p, Select[Divisors[n], PrimeQ]}]; Array[a, 51] (* Stefano Spezia, Jul 14 2025 *)

a(A000040(n)) = 1.
G.f.: Sum_{k>=1} x^prime(k) * (1 + 11*x^prime(k) + 11*x^(2*prime(k)) + x^(3*prime(k))) / (1 - x^prime(k))^5. - Ilya Gutkovskiy, Feb 05 2022
Dirichlet g.f.: zeta(s-4)*primezeta(s). This follows because Sum_{n>=1} a(n)/n^s = Sum_{n>=1} (n^4/n^s) Sum_{p|n} 1/p^4. Since n = p*j, rewrite the sum as Sum_{p} Sum_{j>=1} 1/(p^4*(p*j)^(s-4)) = Sum_{p} 1/p^s Sum_{j>=1} 1/j^(s-4) = zeta(s-4)*primezeta(s). The result generalizes to higher powers of p. - Michael Shamos, Mar 02 2023
Sum_{k=1..n} a(k) ~ A085965 * n^5/5. - Vaclav Kotesovec, Mar 03 2023
a(n) = Sum_{d|n} A059377(d)*A001221(n/d). - Ridouane Oudra, Jul 14 2025
From Wesley Ivan Hurt, Jul 15 2025: (Start)
a(n) = Sum_{d|n} c(d) * (n/d)^4, where c = A010051.
a(p^k) = p^(4*k-4) for p prime and k>=1. (End)

A351245 a(n) = n^5 * Sum_{p|n, p prime} 1/p^5.

Original entry on oeis.org

0, 1, 1, 32, 1, 275, 1, 1024, 243, 3157, 1, 8800, 1, 16839, 3368, 32768, 1, 66825, 1, 101024, 17050, 161083, 1, 281600, 3125, 371325, 59049, 538848, 1, 867151, 1, 1048576, 161294, 1419889, 19932, 2138400, 1, 2476131, 371536, 3232768, 1, 4629701, 1, 5154656, 818424, 6436375, 1

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

nonn

Dirichlet convolution of A010051(n) and n^5. - Wesley Ivan Hurt, Jul 15 2025

			a(6) = 275; a(6) = 6^5 * Sum_{p|6, p prime} 1/p^5 = 7776 * (1/2^5 + 1/3^5) = 275.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k = 0..10: A001221 (k=0), A069359 (k=1), A322078 (k=2), A351242 (k=3), A351244 (k=4), this sequence (k=5), A351246 (k=6), A351247 (k=7), A351248 (k=8), A351249 (k=9), A351262 (k=10).

Cf. A000040, A001221, A010051, A059378, A085966.

Array[#^5*DivisorSum[#, 1/#^5 &, PrimeQ] &, 47] (* Stefano Spezia, Jul 15 2025 *)

a(n) = my(f = factor(n)); sum(i = 1, #f~, (n/f[i,1])^5) \\ David A. Corneth, Jul 15 2025

a(A000040(n)) = 1.
Dirichlet g.f.: zeta(s-5)*primezeta(s). This follows because Sum_{n>=1} a(n)/n^s = Sum_{n>=1} (n^5/n^s) Sum_{p|n} 1/p^5. Since n = p*j, rewrite the sum as Sum_{p} Sum_{j>=1} 1/(p^5*(p*j)^(s-5)) = Sum_{p} 1/p^s Sum_{j>=1} 1/j^(s-5) = zeta(s-5)*primezeta(s). The result generalizes to higher powers of p. - Michael Shamos, Mar 03 2023
Sum_{k=1..n} a(k) ~ A085966 * n^6/6. - Vaclav Kotesovec, Mar 03 2023
a(n) = Sum_{d|n} A059378(d)*A001221(n/d). - Ridouane Oudra, Jul 14 2025
From Wesley Ivan Hurt, Jul 15 2025: (Start)
a(n) = Sum_{d|n} c(d) * (n/d)^5, where c = A010051.
a(p^k) = p^(5*k-5) for p prime and k>=1. (End)

A351246 a(n) = n^6 * Sum_{p|n, p prime} 1/p^6.

Original entry on oeis.org

0, 1, 1, 64, 1, 793, 1, 4096, 729, 15689, 1, 50752, 1, 117713, 16354, 262144, 1, 578097, 1, 1004096, 118378, 1771625, 1, 3248128, 15625, 4826873, 531441, 7533632, 1, 12437281, 1, 16777216, 1772290, 24137633, 133274, 36998208, 1, 47045945, 4827538, 64262144, 1, 93342313

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

nonn

Dirichlet convolution of A010051(n) and n^6. - Wesley Ivan Hurt, Jul 15 2025

			a(6) = 793; a(6) = 6^6 * Sum_{p|6, p prime} 1/p^6 = 46656 * (1/2^6 + 1/3^6) = 793.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k = 0..10: A001221 (k=0), A069359 (k=1), A322078 (k=2), A351242 (k=3), A351244 (k=4), A351245 (k=5), this sequence (k=6), A351247 (k=7), A351248 (k=8), A351249 (k=9), A351262 (k=10).

Cf. A000040, A001221, A010051, A069091.

Array[#^6*DivisorSum[#, 1/#^6 &, PrimeQ] &, 50] (* Wesley Ivan Hurt, Jul 15 2025 *)

a(A000040(n)) = 1.
a(n) = Sum_{d|n} A069091(d)*A001221(n/d). - Ridouane Oudra, Jul 14 2025
From Wesley Ivan Hurt, Jul 15 2025: (Start)
a(n) = Sum_{d|n} c(d) * (n/d)^6, where c = A010051.
a(p^k) = p^(6*k-6) for p prime and k>=1. (End)

A351248 a(n) = n^8 * Sum_{p|n, p prime} 1/p^8.

Original entry on oeis.org

0, 1, 1, 256, 1, 6817, 1, 65536, 6561, 390881, 1, 1745152, 1, 5765057, 397186, 16777216, 1, 44726337, 1, 100065536, 5771362, 214359137, 1, 446758912, 390625, 815730977, 43046721, 1475854592, 1, 2664570241, 1, 4294967296, 214365442, 6975757697, 6155426, 11449942272

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

nonn

Dirichlet convolution of A010051(n) and n^8. - Wesley Ivan Hurt, Jul 15 2025

			a(6) = 6817; a(6) = 6^8 * Sum_{p|6, p prime} 1/p^8 = 1679616 * (1/2^8 + 1/3^8) = 6817.

Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k = 0..10: A001221 (k=0), A069359 (k=1), A322078 (k=2), A351242 (k=3), A351244 (k=4), A351245 (k=5), A351246 (k=6), A351247 (k=7), this sequence (k=8), A351249 (k=9), A351262 (k=10).

Cf. A000040, A001221, A010051, A069093.

Array[#^8*DivisorSum[#, 1/#^8 &, PrimeQ] &, 36] (* Stefano Spezia, Jul 15 2025 *)

from sympy import primefactors
def A351248(n): return sum((n//p)**8 for p in primefactors(n)) # Chai Wah Wu, Feb 05 2022

a(A000040(n)) = 1.
a(n) = Sum_{d|n} A069093(d)*A001221(n/d). - Ridouane Oudra, Jul 15 2025
From Wesley Ivan Hurt, Jul 15 2025: (Start)
a(n) = Sum_{d|n} c(d) * (n/d)^8, where c = A010051.
a(p^k) = p^(8*k-8) for p prime and k>=1. (End)

A351249 a(n) = n^9 * Sum_{p|n, p prime} 1/p^9.

Original entry on oeis.org

0, 1, 1, 512, 1, 20195, 1, 262144, 19683, 1953637, 1, 10339840, 1, 40354119, 1972808, 134217728, 1, 397498185, 1, 1000262144, 40373290, 2357948203, 1, 5293998080, 1953125, 10604499885, 387420489, 20661308928, 1, 39453437071, 1, 68719476736, 2357967374, 118587877009, 42306732

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

nonn

Dirichlet convolution of A010051(n) and n^9. - Wesley Ivan Hurt, Jul 15 2025

			a(6) = 20195; a(6) = 6^9 * Sum_{p|6, p prime} 1/p^9 = 10077696 * (1/2^9 + 1/3^9) = 20195.

Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k = 0..10: A001221 (k=0), A069359 (k=1), A322078 (k=2), A351242 (k=3), A351244 (k=4), A351245 (k=5), A351246 (k=6), A351247 (k=7), A351248 (k=8), this sequence (k=9), A351262 (k=10).

Cf. A000040, A001221, A010051, A069094.

Array[#^9*DivisorSum[#, 1/#^9 &, PrimeQ] &, 50] (* Wesley Ivan Hurt, Jul 15 2025 *)

from sympy import primefactors
def A351249(n): return sum((n//p)**9 for p in primefactors(n)) # Chai Wah Wu, Feb 05 2022

a(A000040(n)) = 1.
a(n) = Sum_{d|n} A069094(d)*A001221(n/d). - Ridouane Oudra, Jul 15 2025
From Wesley Ivan Hurt, Jul 15 2025: (Start)
a(n) = Sum_{d|n} c(d) * (n/d)^9, where c = A010051.
a(p^k) = p^(9*k-9) for p prime and k>=1. (End)

A351262 a(n) = n^10 * Sum_{p|n, p prime} 1/p^10.

Original entry on oeis.org

0, 1, 1, 1024, 1, 60073, 1, 1048576, 59049, 9766649, 1, 61514752, 1, 282476273, 9824674, 1073741824, 1, 3547250577, 1, 10001048576, 282534298, 25937425625, 1, 62991106048, 9765625, 137858492873, 3486784401, 289255703552, 1, 586710856801, 1, 1099511627776, 25937483650

Offset: 1

Wesley Ivan Hurt, Feb 05 2022

nonn

Dirichlet convolution of A010051(n) and n^10. - Wesley Ivan Hurt, Jul 15 2025

			a(6) = 60073; a(6) = 6^10 * Sum_{p|6, p prime} 1/p^10 = 60466176 * (1/2^10 + 1/3^10) = 60073.

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

Sequences of the form n^k * Sum_{p|n, p prime} 1/p^k for k = 0..10: A001221 (k=0), A069359 (k=1), A322078 (k=2), A351242 (k=3), A351244 (k=4), A351245 (k=5), A351246 (k=6), A351247 (k=7), A351248 (k=8), A351249 (k=9), this sequence (k=10).

Cf. A000040, A001221, A010051, A069095.

f:= proc(n) local p;
  n^10 * add(1/p^10, p = numtheory:-factorset(n))
end proc:
map(f, [$1..40]); # Robert Israel, Sep 10 2024

Join[{0},Table[n^10 Total[1/FactorInteger[n][[;;,1]]^10],{n,2,40}]] (* Harvey P. Dale, Aug 10 2024 *)

a(n) = my(f=factor(n)); n^10*sum(k=1, #f~, 1/f[k,1]^10); \\ Michel Marcus, Sep 10 2024

from sympy import primefactors
def A351262(n): return sum((n//p)**10 for p in primefactors(n)) # Chai Wah Wu, Feb 05 2022

a(A000040(n)) = 1.
a(n) = Sum_{d|n} A069095(d)*A001221(n/d). - Ridouane Oudra, Jul 15 2025
From Wesley Ivan Hurt, Jul 15 2025: (Start)
a(n) = Sum_{d|n} c(d) * (n/d)^10, where c = A010051.
a(p^k) = p^(10*k-10) for p prime and k>=1. (End)

cp's OEIS Frontend

A001221 Number of distinct primes dividing n (also called omega(n)).

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A069359 a(n) = n * Sum_{p|n} 1/p where p are primes dividing n.

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A322078 a(n) = n^2 * Sum_{p|n, p prime} 1/p^2.

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A351242 a(n) = n^3 * Sum_{p|n, p prime} 1/p^3.

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A351244 a(n) = n^4 * Sum_{p|n, p prime} 1/p^4.

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A351245 a(n) = n^5 * Sum_{p|n, p prime} 1/p^5.

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