A173557 -id:A173557 - cp's OEIS frontend

A345001 a(n) = sigma(n) + n' - 2n, where n' is the arithmetic derivative of n (A003415) and sigma is the sum of divisors (A000203).

-1, 0, -1, 3, -3, 5, -5, 11, 1, 5, -9, 20, -11, 5, 2, 31, -15, 24, -17, 26, 0, 5, -21, 56, -9, 5, 13, 32, -27, 43, -29, 79, -4, 5, -10, 79, -35, 5, -6, 78, -39, 53, -41, 44, 27, 5, -45, 140, -27, 38, -10, 50, -51, 93, -22, 100, -12, 5, -57, 140, -59, 5, 29, 191, -28, 73, -65, 62, -16, 63, -69, 207, -71, 5, 29, 68

Offset: 1

Antti Karttunen, Jun 05 2021

Coincides with A003415 only on perfect numbers (A000396).

Cf. A000203, A000396, A001065, A003415, A033879, A168036, A344999, A345002, A345003, A345004, A345005, A345049.
Cf. also A051709, A344753 (analogous sequences).
Cf. A013661, A136141.

A003415[n_] := If[n < 2, 0, Module[{f = FactorInteger[n]}, If[PrimeQ[n], 1, Total[n*f[[All, 2]]/f[[All, 1]]]]]];
a[n_] := DivisorSigma[1, n] + A003415[n] - 2 n;
Array[a, 80] (* Jean-François Alcover, Jun 12 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A345001(n) = (sigma(n)+A003415(n)-(2*n));

a(n) = A003415(n) - A033879(n) = A000203(n) + A003415(n) - 2*n.
a(n) = A001065(n) + A168036(n).
a(n) = A344999(n) / A048250(n) = A345049(n) / A173557(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A013661 + A136141 - 2 = 0.418090735898... . - Amiram Eldar, Dec 08 2023

A322360 Multiplicative with a(p^e) = p^2 - 1.

Original entry on oeis.org

1, 3, 8, 3, 24, 24, 48, 3, 8, 72, 120, 24, 168, 144, 192, 3, 288, 24, 360, 72, 384, 360, 528, 24, 24, 504, 8, 144, 840, 576, 960, 3, 960, 864, 1152, 24, 1368, 1080, 1344, 72, 1680, 1152, 1848, 360, 192, 1584, 2208, 24, 48, 72, 2304, 504, 2808, 24, 2880, 144, 2880, 2520, 3480, 576, 3720, 2880, 384, 3, 4032, 2880

Offset: 1

Antti Karttunen, Dec 04 2018

Absolute values of A046970, the Dirichlet inverse of the Jordan function J_2 (A007434).

Absolute values of the Möbius transform of A055491. (See Benoit Cloitre's May 31 2002 comment in A046970).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Absolute values of A046970.

Cf. A048250, A173557, A066086, A322359, A351265.

a:= n-> mul(i[1]^2-1, i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Jan 05 2021

a[n_] := If[n==1, 1, Times @@ ((#^2-1)& @@@ FactorInteger[n])]; Array[a, 50]  (* Amiram Eldar, Dec 05 2018 *)

A322360(n) = factorback(apply(p -> (p*p)-1, factor(n)[, 1]));

A322360(n) = abs(sumdiv(n,d,moebius(n/d)*(core(d)^2)));

Multiplicative with a(p^e) = p^2 - 1.
a(n) = Product_{p prime divides n} (p^2 - 1).
a(n) = abs(A046970(n)).
a(n) = A048250(n) * A173557(n) = A066086(n) * A322359(n).
G.f. for a signed version of the sequence: Sum_{n >= 1} mu(n)*n^2*x^n/(1 - x^n) = Sum_{n >= 1} (-1)^omega(n)*a(n)*x^n = x - 3*x^2 - 8*x^3 - 3*x^4 - 24*x^5 + 24*x^6 - 48*x^7 - 3*x^8 - 8*x^9 + 72*x^10 - ..., where mu(n) is the Möbius function A008683(n) and omega(n) = A001221(n) is the number of distinct primes dividing n. - Peter Bala, Mar 05 2022
Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime} (p-1)*(p^2 + 2*p + 2)/(p*(p^2 + p + 1)) = 0.187556464... . - Amiram Eldar, Oct 22 2022
a(n) = A007434(A007947(n)). - Enrique Pérez Herrero, Oct 14 2024

A326297 If n = Product (p_j^k_j) then a(n) = Product ((p_j - 1)^(k_j - 1)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 4, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 4, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 6, 2, 4

Offset: 1

Ilya Gutkovskiy, Mar 03 2020

			a(98) = a(2 * 7^2) = (2 - 1)^(1 - 1) * (7 - 1)^(2 - 1) = 6.

Cf. A003557, A003958, A003959, A007947, A023900, A064478, A064549, A122132 (positions of 1's), A125131, A173557, A325126, A327564.

seq(mul((p-1)^(padic[ordp](n,p)-1), p in numtheory[factorset](n)), n =1..100); # Ridouane Oudra, Oct 29 2024

a[n_] := If[n == 1, 1, Times @@ ((#[[1]] - 1)^(#[[2]] - 1) & /@ FactorInteger[n])]; Table[a[n], {n, 1, 100}]

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1]--; f[k,2]--); factorback(f); \\ Michel Marcus, Mar 03 2020

from math import prod
from sympy import factorint
def a(n): return prod((p-1)**(e-1) for p, e in factorint(n).items())
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Aug 30 2021

a(n) = A003958(n) / abs(A023900(n)) = abs(A325126(n)) / A007947(n).
Dirichlet g.f.: Product_{p prime} (1 + 1/(p^s - p + 1)). - Amiram Eldar, Dec 07 2023
a(n) = A003958(n)/A173557(n). - Ridouane Oudra, Oct 29 2024

A299822 Product of Euler's totient and the squarefree kernel, a(n) = phi(n)*rad(n).

Original entry on oeis.org

1, 2, 6, 4, 20, 12, 42, 8, 18, 40, 110, 24, 156, 84, 120, 16, 272, 36, 342, 80, 252, 220, 506, 48, 100, 312, 54, 168, 812, 240, 930, 32, 660, 544, 840, 72, 1332, 684, 936, 160, 1640, 504, 1806, 440, 360, 1012, 2162, 96, 294, 200, 1632, 624, 2756, 108, 2200, 336, 2052, 1624

Offset: 1

R. J. Mathar, Feb 19 2018

A permutation of A323333. - Amiram Eldar, Sep 19 2020

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A007947, A065485, A173557, A323333.

A299822 := proc(n)
    local a,p,e,pe;
    a := 1;
    for pe in ifactors(n)[2] do
        p := pe[1] ; e:= pe[2] ;
        a := a*p*(p-1)*p^(e-1) ;
    end do:
    a ;
end proc:
seq(A299822(n),n=1..130) ;

Array[EulerPhi[#] SelectFirst[Reverse@ Divisors@ #, SquareFreeQ] &, 58] (* Michael De Vlieger, Feb 20 2018 *)
f[p_, e_] := (p-1)*p^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2020 *)

a(n) = eulerphi(n)*factorback(factorint(n)[, 1]); \\ Michel Marcus, Jun 24 2019

a(n) = A000010(n)*A007947(n) = n*A173557(n).
Dirichlet g.f.: zeta(s-1)*Product_{p prime} (1 - 2*p^(1-s) + p^(2-s)), corrected by Vaclav Kotesovec, Dec 18 2019
Multiplicative with a(p^e) = p*(p-1)*p^(e-1).
a(n) = n*abs(A023900(n)). (Trivially rephrasing a formula in A173557.) - Omar E. Pol, Feb 19 2018
a(2^e) = 2^e. (Special case of above.) - Omar E. Pol, Feb 19 2018
A003557(n) | a(n). - R. J. Mathar, Feb 26 2018
From Vaclav Kotesovec, Dec 18 2019: (Start)
Dirichlet g.f.: zeta(s-1) * zeta(s-2) * Product_{primes p} (1 + 2*p^(3-2*s) - p^(4-2*s) - 2*p^(1-s)).
Sum_{k=1..n} a(k) ~ c * Pi^2 * n^3 / 18, where c = A065473 = Product_{primes p} (1 - 3/p^2 + 2/p^3) = 0.2867474284344787341078927... (End)
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p-1)^2) = 2.826419... (A065485). - Amiram Eldar, Sep 19 2020
G.f. for a signed version of the sequence: Sum_{n >= 1} mu(n)*n^2*x^n/(1 - x^n)^2 = Sum_{n >= 1} (-1)^omega(n)*a(n)*x^n = x - 2*x^2 - 6*x^3 - 4*x^4 - 20*x^5 + 12*x^6 - 42*x^7 - 8*x^8 - 18*x^9 + 40*x^10 - ..., where mu(n) is the Möbius function A008683(n) and omega(n) = A001221(n) is the number of distinct primes dividing n. - Peter Bala, Mar 05 2022

A304408 If n = Product (p_j^k_j) then a(n) = Product ((p_j - 1)*(k_j + 1)).

Original entry on oeis.org

1, 2, 4, 3, 8, 8, 12, 4, 6, 16, 20, 12, 24, 24, 32, 5, 32, 12, 36, 24, 48, 40, 44, 16, 12, 48, 8, 36, 56, 64, 60, 6, 80, 64, 96, 18, 72, 72, 96, 32, 80, 96, 84, 60, 48, 88, 92, 20, 18, 24, 128, 72, 104, 16, 160, 48, 144, 112, 116, 96, 120, 120, 72, 7, 192, 160, 132, 96, 176, 192

Offset: 1

Ilya Gutkovskiy, May 12 2018

			a(20) = a(2^2*5) = (2 - 1)*(2 + 1) * (5 - 1)*(1 + 1) = 24.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000
Index entries for sequences computed from indices in prime factorization
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A000010, A000026, A000040, A000302 (numbers n such that a(n) is odd), A001221, A006093, A007947, A023900, A034444, A059975, A062355, A173557, A304407, A304409, A304411, A304412.

a:= n-> mul((i[1]-1)*(i[2]+1), i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Jan 05 2021

a[n_] := Times @@ ((#[[1]] - 1) (#[[2]] + 1) & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 70}]
Table[DivisorSigma[0, n] EulerPhi[Last[Select[Divisors[n], SquareFreeQ]]], {n, 70}]

a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); (p-1)*(e+1))} \\ Andrew Howroyd, Jul 24 2018

a(n) = A000005(n)*abs(A023900(n)) = A000005(n)*A173557(n) = A000005(n)*A000010(A007947(n)).
a(p^k) = (p - 1)*(k + 1) where p is a prime and k > 0.
a(n) = 2^omega(n)*phi(n) if n is a squarefree (A005117), where omega() = A001221 and phi() = A000010.

A335341 Sum of divisors of A003557(n).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 7, 4, 1, 1, 3, 1, 1, 1, 15, 1, 4, 1, 3, 1, 1, 1, 7, 6, 1, 13, 3, 1, 1, 1, 31, 1, 1, 1, 12, 1, 1, 1, 7, 1, 1, 1, 3, 4, 1, 1, 15, 8, 6, 1, 3, 1, 13, 1, 7, 1, 1, 1, 3, 1, 1, 4, 63, 1, 1, 1, 3, 1, 1, 1, 28, 1, 1, 6, 3, 1, 1, 1

Offset: 1

R. J. Mathar, Jun 02 2020

The sum of the divisors d of n such that n/d is a coreful divisor of n (a coreful divisor of n is a divisor with the same squarefree kernel as n). The number of these divisors is A005361(n). - Amiram Eldar, Jun 30 2023

Antti Karttunen, Table of n, a(n) for n = 1..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000203, A003557, A005361 (number of divisors of A003557), A336567.

Cf. A047994, A173557.

A335341 := proc(n)
    local a,pe,p,e ;
    a := 1;
    for pe in ifactors(n)[2] do
        p := op(1,pe) ;
        e := op(2,pe) ;
        if e > 1 then
            a := a*(p^e-1)/(p-1) ;
        end if;
    end do:
    a ;
end proc:

f[p_, e_] := (p^e-1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 26 2020 *)

a(n) = sigma(n/factorback(factor(n)[, 1])); \\ Michel Marcus, Jun 02 2020

a(n) = A000203(A003557(n)).
Multiplicative with a(p^1)=1 and a(p^e) = (p^e-1)/(p-1) if e>1.
A057723(n) = A007947(n)*a(n).
a(n) = 1 iff n in A005117.
a(n) = A336567(n) + A003557(n). - Antti Karttunen, Jul 28 2020
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 1/p^(s-1) + 1/p^(2*s-1)). - Amiram Eldar, Sep 09 2023
a(n) = A047994(n)/A173557(n). - Ridouane Oudra, Oct 30 2023

A345051 Numbers k such that A345048(k) is equal to A345049(k).

Original entry on oeis.org

2, 6, 9, 15, 28, 496, 625, 1225, 3993, 8128, 117649, 218491, 857375, 3788435, 4259571, 33550336, 69302975, 136410197, 200533921, 313742585, 603439225, 1516358753, 2563893625, 3326174929, 5655792025, 8589869056, 10214476341

Offset: 1

Antti Karttunen, Jun 06 2021

Numbers k for which A342001(n) * A051709(n) = A173557(n) * A345001(n).

Conjecture: Sequence is a disjoint union of A000396 and A166374, i.e., there are no terms of any other kind.

Index entries for sequences where odd perfect numbers must occur, if they exist at all.

Cf. A051709, A173557, A342001, A345001, A345048, A345049.
Positions of zeros in A345050.
Cf. A000396, A166374 (subsequences).
Cf. also A345003.

PARI
```
isA345051(n) = (0==A345050(n));
```

a(21)-a(27) from Amiram Eldar, Dec 08 2023

A068997 Numbers k such that Sum_{d|k} d*mu(d) divides k.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 32, 36, 40, 48, 54, 64, 72, 80, 84, 96, 100, 108, 120, 128, 144, 160, 162, 168, 192, 200, 216, 240, 252, 256, 272, 288, 312, 320, 324, 336, 360, 384, 400, 432, 440, 480, 486, 500, 504, 512, 544, 576, 588, 600, 624, 640, 648

Offset: 1

Benoit Cloitre, Apr 07 2002

Numbers k such that A023900(k) divides k.
The only squarefree terms so far are a(1), a(2), and a(4). - Torlach Rush, Dec 04 2017
There are no more squarefree terms. The squarefree terms are also the squarefree terms of A007694 since A023900(n) = A008683(n) * A000010(n) for squarefree numbers n, and A007694 contains only 3-smooth numbers (A003586). - Amiram Eldar, Apr 19 2025
There is a surjective mapping from all even numbers not in this sequence to terms of the sequence. The first such is 10 to a(9). The next is 14, 28, 42 to a(19). All even numbers not in the sequence are divisors of some term in the sequence. - Torlach Rush, Dec 08 2017

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A000010, A003586, A007694, A008683, A023900, A173557, A293928.

a068997 n = a068997_list !! (n - 1)
a068997_list = filter (\x -> mod x (a173557 x) == 0) [1..]
-- Reinhard Zumkeller, Jun 01 2015

with(numtheory): A068997 := i->`if`(i mod phi(mul(j,j=factorset(i)))=0,i,NULL): seq(A068997(i),i=1..650); # Peter Luschny, Nov 02 2010

Select[Range[650], Divisible[#, DivisorSum[#, # MoebiusMu[#] &]] &] (* Michael De Vlieger, Nov 20 2017 *)
q[1] =True; q[n_] := Divisible[n, Times @@ ((First[#] - 1) & /@ FactorInteger[n])]; Select[Range[650], q] (* Amiram Eldar, Apr 19 2025 *)

for(n=1,1000,if(n%sumdiv(n,d,moebius(d)*d)==0,print1(n,",")))

isok(k) = !(k % vecprod(apply(x -> 1-x, factor(k)[, 1]))); \\ Amiram Eldar, Apr 19 2025

A304407 If n = Product (p_j^k_j) then a(n) = Product ((p_j - 1)*k_j).

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 3, 4, 4, 10, 4, 12, 6, 8, 4, 16, 4, 18, 8, 12, 10, 22, 6, 8, 12, 6, 12, 28, 8, 30, 5, 20, 16, 24, 8, 36, 18, 24, 12, 40, 12, 42, 20, 16, 22, 46, 8, 12, 8, 32, 24, 52, 6, 40, 18, 36, 28, 58, 16, 60, 30, 24, 6, 48, 20, 66, 32, 44, 24, 70, 12, 72, 36, 16

Offset: 1

Ilya Gutkovskiy, May 12 2018

			a(60) = a(2^2*3*5) = (2 - 1)*2 * (3 - 1)*1 * (5 - 1)*1 = 16.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000.
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences computed from indices in prime factorization.

Cf. A000010, A000026, A000040, A002110, A003958, A005117, A005361, A005867, A006093, A007947, A023900, A048250, A059975, A068997, A081294 (numbers n such that a(n) is odd), A087656, A090624, A152649, A173557, A304117, A304408, A304409, A304411, A304412.

Cf. A001221, A076479.

seq(mul((p-1)*padic[ordp](n, p), p in numtheory[factorset](n)), n=1..100); # Ridouane Oudra, Jun 06 2025

a[n_] := Times @@ ((#[[1]] - 1) #[[2]] & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 75}]
Table[EulerPhi[Last[Select[Divisors[n], SquareFreeQ]]] DivisorSigma[0, n/Last[Select[Divisors[n], SquareFreeQ]]], {n, 75}]

a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); (p-1)*e)} \\ Andrew Howroyd, Jul 24 2018

a(n) = A005361(n)*abs(A023900(n)) = A005361(n)*A173557(n) = A005361(n)*A000010(A007947(n)).
a(p^k) = (p - 1)*k where p is a prime and k > 0.
a(n) = phi(n) if n is a squarefree (A005117), where phi() = A000010.
a(A002110(k)) = A005867(k).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^4/72) * Product_{p prime} (1 - 4/p^2 + 3/p^3 + 1/p^4 - 1/p^5) = 0.2644703894... . - Amiram Eldar, Nov 30 2022
a(n) = (-1)^A001221(n) * (Sum_{d1|n} Sum_{d2|n} mu(d1)*gcd(d1,d2)). - Ridouane Oudra, Jun 06 2025

A305444 a(n) = Product_{p is odd and prime and divisor of n} (p - 2).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 5, 1, 1, 3, 9, 1, 11, 5, 3, 1, 15, 1, 17, 3, 5, 9, 21, 1, 3, 11, 1, 5, 27, 3, 29, 1, 9, 15, 15, 1, 35, 17, 11, 3, 39, 5, 41, 9, 3, 21, 45, 1, 5, 3, 15, 11, 51, 1, 27, 5, 17, 27, 57, 3, 59, 29, 5, 1, 33, 9, 65, 15, 21, 15, 69, 1, 71, 35, 3, 17

Offset: 1

Markus Sigg, Aug 12 2018

Denominator of c_n = Product_{odd p| n} (p-1)/(p-2). Numerator is A173557. [Yamasaki and Yamasaki]. - N. J. A. Sloane, Jan 19 2020

This ratio, multiplied by the twin prime constant, occurs in the asymptotic behavior of prime gaps of size 2*n as decribed by the Hardy-Littlewood asymptotic conjecture for the number of prime pairs. See A005597 for more information. - Hugo Pfoertner, Dec 25 2024

Markus Sigg, Table of n, a(n) for n = 1..10000
Hugo Pfoertner, Plot of A173557(n)/a(n) vs n, using Plot 2.
Yasuo Yamasaki and Aiichi Yamasaki, On the Gap Distribution of Prime Numbers, Kyoto University Research Information Repository, October 1994. MR1370273 (97a:11141).

Cf. A173557, A000010, A008683, A001221.

A305444 := proc(n) mul(d - 2, d = numtheory[factorset](n) minus {2}) end proc:

a[n_] := If[n == 1, 1, Times @@ (DeleteCases[FactorInteger[n][[All, 1]], 2] - 2)];
Array[a, 100] (* Jean-François Alcover, Apr 08 2020*)

a(n)={my(f=factor(n>>valuation(n,2))[,1]); prod(i=1, #f, f[i]-2)} \\ Andrew Howroyd, Aug 12 2018

from math import prod
from sympy import primefactors
def A305444(n): return prod(p-2 for p in primefactors(n>>(~n&n-1).bit_length())) # Chai Wah Wu, Sep 08 2023

Sum_{k=1..n} a(k) ~ c * n^2, where c = (2/3) * Product_{p prime} (1 - 3/(p*(p+1))) = 0.1950799046... . - Amiram Eldar, Nov 12 2022
a(n) = abs( Sum_{d divides n, d odd} mobius(d) * phi(d) ). - Peter Bala, Feb 01 2024
a(n) = (-1)^omega(n) * Sum_{d|n} mu(d)*phi(2*d), where omega = A001221. - Ridouane Oudra, Jul 30 2025

cp's OEIS Frontend

A345001 a(n) = sigma(n) + n' - 2n, where n' is the arithmetic derivative of n (A003415) and sigma is the sum of divisors (A000203).

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A322360 Multiplicative with a(p^e) = p^2 - 1.

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A326297 If n = Product (p_j^k_j) then a(n) = Product ((p_j - 1)^(k_j - 1)).

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A299822 Product of Euler's totient and the squarefree kernel, a(n) = phi(n)*rad(n).

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A304408 If n = Product (p_j^k_j) then a(n) = Product ((p_j - 1)*(k_j + 1)).

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A335341 Sum of divisors of A003557(n).

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A345051 Numbers k such that A345048(k) is equal to A345049(k).

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