A064549 -id:A064549 - cp's OEIS frontend

A280286 a(n) is the least k such that sopfr(k) - sopf(k) = n.

4, 9, 8, 25, 16, 49, 32, 81, 64, 121, 128, 169, 256, 625, 512, 289, 1024, 361, 2048, 1444, 1331, 529, 5324, 2116, 2197, 4232, 8788, 841, 17576, 961, 7569, 3844, 4913, 7688, 19652, 1369, 6859, 5476, 12321, 1681, 34225, 1849, 15129, 7396, 12167, 2209, 46225, 8836, 19881

Offset: 2

Michel Marcus, Dec 31 2016

nonn

Michel Marcus, Table of n, a(n) for n = 2..500

Cf. A001414 (sopfr), A008472 (sopf), A001248, A280163.
A multiplicative version is A064549 (sorted A001694), firsts of A003557.
For length instead of sum we have A151821.
These are the positions of first appearances in A280292 = A001414 - A008472.
For indices instead of factors we have A380956 (sorted A380957), firsts of A380955.
A multiplicative version for indices is A380987 (sorted A380988), firsts of A290106.
For prime exponents instead of factors we have A380989, firsts of A380958.
The sorted version is A381075.
For product instead of sum see A381076, sorted firsts of A066503.
A000040 lists the primes, differences A001223.
A005117 lists squarefree numbers, complement A013929.
A020639 gives least prime factor (index A055396), greatest A061395 (index A006530).
A056239 adds up prime indices, row sums of A112798.
Cf. A000720, A001222, A046660, A071625, A075255, A116861, A136565, A156061, A178503, A175508, A364916, A366528.

prifacs[n_]:=If[n==1,{},Flatten[Apply[ConstantArray,FactorInteger[n],{1}]]];
q=Table[Total[prifacs[n]]-Total[Union[prifacs[n]]],{n,1000}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
Table[Position[q,k][[1,1]],{k,2,mnrm[q/.(0->1)]}] (* Gus Wiseman, Feb 20 2025 *)

sopfr(n) = my(f=factor(n)); sum(j=1, #f~, f[j,1]*f[j,2]);
sopf(n) = my(f=factor(n)); sum(j=1, #f~, f[j,1]);
a(n) = {my(k = 2); while (sopfr(k) - sopf(k) != n, k++); k;}

For p prime, a(p) = p^2 (see A001248).

A356191 a(n) is the smallest exponentially odd number that is divisible by n.

Original entry on oeis.org

1, 2, 3, 8, 5, 6, 7, 8, 27, 10, 11, 24, 13, 14, 15, 32, 17, 54, 19, 40, 21, 22, 23, 24, 125, 26, 27, 56, 29, 30, 31, 32, 33, 34, 35, 216, 37, 38, 39, 40, 41, 42, 43, 88, 135, 46, 47, 96, 343, 250, 51, 104, 53, 54, 55, 56, 57, 58, 59, 120, 61, 62, 189, 128, 65

Offset: 1

Amiram Eldar, Jul 29 2022

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

Cf. A064549, A007913, A268335, A336643, A350390.

Similar sequences: A053149, A053143, A053167, A066638, A087320, A087321, A197863, A356192, A356193, A356194.

f[p_, e_] := If[OddQ[e], p^e, p^(e + 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f=factor(n)); prod(i=1, #f~, if(f[i,2]%2, f[i,1]^f[i,2], f[i,1]^(f[i,2]+1)))};

for(n=1, 100, print1(direuler(p=2, n, 1/(1 - p^2*X^2) * (1 + p*X + p^3*X^2 - p^2*X^2))[n], ", ")) \\ Vaclav Kotesovec, Sep 09 2023

Multiplicative with a(p^e) = p^e if e is odd and p^(e+1) otherwise.
a(n) = n iff n is in A268335.
a(n) = A064549(n)/A007913(n).
a(n) = n*A336643(n).
a(n) = n^2/A350390(n).
From Vaclav Kotesovec, Sep 09 2023: (Start)
Let f(s) = Product_{p prime} (1 - p^(6-5*s) + p^(7-5*s) + 2*p^(5-4*s) - p^(6-4*s) + p^(3-3*s) - p^(4-3*s) - 2*p^(2-2*s)).
Sum_{k=1..n} a(k) ~ Pi^2 * f(2) * n^2 / 24 * (log(n) + 3*gamma - 1/2 + 12*zeta'(2)/Pi^2 + f'(2)/f(2)), where
f(2) = Product_{p prime} (1 - 4/p^2 + 4/p^3 - 1/p^4) = A256392 = 0.2177787166195363783230075141194468131307977550013559376482764035236264911...,
f'(2) = f(2) * Sum_{p prime} (11*p - 5) * log(p) / (p^3 + p^2 - 3*p + 1) = f(1) * 4.7165968208567630786609552448708126340725121316268495170070986645608062483...
and gamma is the Euler-Mascheroni constant A001620. (End)

A327564 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, 3, 1, 1, 1, 9, 4, 1, 1, 3, 1, 1, 1, 27, 1, 4, 1, 3, 1, 1, 1, 9, 6, 1, 16, 3, 1, 1, 1, 81, 1, 1, 1, 12, 1, 1, 1, 9, 1, 1, 1, 3, 4, 1, 1, 27, 8, 6, 1, 3, 1, 16, 1, 9, 1, 1, 1, 3, 1, 1, 4, 243, 1, 1, 1, 3, 1, 1, 1, 36, 1, 1, 6, 3, 1, 1, 1, 27, 64, 1, 1, 3, 1

Offset: 1

Ilya Gutkovskiy, Mar 03 2020

			a(12) = a(2^2 * 3) = (2 + 1)^(2 - 1) * (3 + 1)^(1 - 1) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences computed from indices in prime factorization.

Cf. A001221, A003557, A003958, A003959, A003968, A005117 (positions of 1's), A007947, A048250, A064478, A064549, A173557, A323363, A325126, A326297, A340368, A348038, A348039, A347960.

a[n_] := Times @@ ((#[[1]] + 1)^(#[[2]] - 1) & /@ FactorInteger[n]); Table[a[n], {n, 1, 85}]

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

a(1) = 1; a(n) = -Sum_{d|n, dA001221(n/d) * A003557(n/d) * a(d).
a(n) = A003959(n) / A048250(n) = A003968(n) / A007947(n).
a(n) = A348038(n) * A348039(n) = A340368(n) / A173557(n). - Antti Karttunen, Oct 29 2021
Dirichlet g.f.: 1/(zeta(s-1) * Product_{p prime} (1 - 1/p^(s-1) - 1/p^s)). - Amiram Eldar, Dec 07 2023

A073353 Sum of n and its squarefree kernel.

Original entry on oeis.org

2, 4, 6, 6, 10, 12, 14, 10, 12, 20, 22, 18, 26, 28, 30, 18, 34, 24, 38, 30, 42, 44, 46, 30, 30, 52, 30, 42, 58, 60, 62, 34, 66, 68, 70, 42, 74, 76, 78, 50, 82, 84, 86, 66, 60, 92, 94, 54, 56, 60, 102, 78, 106, 60, 110, 70, 114, 116, 118, 90, 122, 124, 84, 66, 130, 132

Offset: 1

Reinhard Zumkeller, Jul 29 2002

a(n) is even; a(n)=2*n iff n is squarefree.
Least k >n such that n divides k^n. - Benoit Cloitre, Oct 09 2002
a(n) is the smallest integer > n such that the positive integers coprime to a(n) are also coprime to n. - Leroy Quet, Dec 24 2006

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

Cf. A007947, A005117, A065463, A066503, A064549, A003557, A073354, A078310.

a073353 n = n + a007947 n  -- Reinhard Zumkeller, Jul 23 2013

a[n_] := n + Times @@ FactorInteger[n][[;;, 1]]; Array[a, 100] (* Amiram Eldar, Dec 07 2023 *)

a(n)=vecprod(factor(n)[,1])+n \\ Charles R Greathouse IV, Nov 05 2017

a(n) = n + A007947(n).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 + A065463 = 1.704442... . - Amiram Eldar, Dec 07 2023

A295294 Sum of the divisors of the powerful part of n: a(n) = A000203(A057521(n)).

Original entry on oeis.org

1, 1, 1, 7, 1, 1, 1, 15, 13, 1, 1, 7, 1, 1, 1, 31, 1, 13, 1, 7, 1, 1, 1, 15, 31, 1, 40, 7, 1, 1, 1, 63, 1, 1, 1, 91, 1, 1, 1, 15, 1, 1, 1, 7, 13, 1, 1, 31, 57, 31, 1, 7, 1, 40, 1, 15, 1, 1, 1, 7, 1, 1, 13, 127, 1, 1, 1, 7, 1, 1, 1, 195, 1, 1, 31, 7, 1, 1, 1, 31, 121, 1, 1, 7, 1, 1, 1, 15, 1, 13, 1, 7, 1, 1, 1, 63

Offset: 1

Antti Karttunen, Nov 25 2017

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

Cf. A000203, A001694, A003557, A005117, A057521, A064549, A092261, A295295.

Array[DivisorSigma[1, #/Denominator[#/Apply[Times, FactorInteger[#][[All, 1]]]^2] ] &, 96] (* Michael De Vlieger, Nov 26 2017, after Jean-François Alcover at A057521 *)
f[p_, e_] := If[e == 1, 1, (p^(e+1)-1)/(p-1)]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 08 2022 *)

a(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i=1, #p, if(e[i] == 1, 1, (p[i]^(e[i]+1)-1)/(p[i]-1)))}; \\ Amiram Eldar, Oct 08 2022

from math import prod
from sympy import factorint
def A295294(n): return prod((p**(e+1)-1)//(p-1) for p, e in factorint(n).items() if e > 1) # Chai Wah Wu, Nov 14 2022

(define (A295294 n) (A000203 (A057521 n)))
;; With memoization-macro definec:
(definec (A295294 n) (if (= 1 n) n (let ((p (A020639 n)) (e (A067029 n))) (* (if (= e 1) 1 (/ (- (expt p (+ 1 e)) 1) (- p 1))) (A295294 (A028234 n))))))

Multiplicative with a(p) = 1 and a(p^e) = (p^(e+1)-1)/(p-1) for e > 1.
a(n) = A000203(A057521(n)) = A000203(A064549(A003557(n))).
a(n) = A000203(n) / A092261(n).
From Amiram Eldar, Oct 08 2022: (Start)
a(n) = 1 iff n is squarefree (A005117).
a(n) = A000203(n) iff n is powerful (A001694). (End)
Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 - 1/p^(s-1) + 1/p^(2*s-2) + 1/p^(2*s-1) - 1/p^(3*s-2)). - Amiram Eldar, Sep 09 2023

A307959 Primitive coreful perfect numbers: powerful numbers k such that csigma(k) = 2*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

Original entry on oeis.org

36, 392, 30752, 2064512, 549621604352, 2251765454077952, 144114638320566272, 9903520305059670164485701632, 12259964326927110856232952250923146488025299504439754752, 237142198758023568227473376531545064850552499416058362196624559881526665860349952

Offset: 1

Amiram Eldar, May 08 2019

nonn

All the coreful perfect numbers (A307958) can be obtained from a primitive term k by multiplying it by m, if m is squarefree and coprime to k. The primitive terms are powerful. If k = m * r is in the sequence where r = rad(k) is the squarefree kernel of k (A007947), then the sum of the coreful divisors of k is csigma(k) = csigma(m * r) = sigma(m) * r, where sigma(m) is the sum of all the divisors of m (A000203). Thus k is a primitive coreful perfect number, iff sigma(m) * r = 2 * m * r, or m = k/rad(k) is a perfect number. Since k is powerful k = m * rad(m), thus all the primitive coreful perfect numbers can be generated from the perfect numbers by multiplying them by their squarefree kernel. The even terms of these sequence are (2^p) * (2^p-1)^2, were p is a Mersenne exponent (A000043). There is an odd term in this sequence iff there is an odd perfect number.

			The first coreful perfect numbers are 36, 180 = 36*5, 252 = 36*7, 392, 396 = 36*11, 468 = 36*13, ... thus the primitive ones are 36, 392, ...

Cf. A000043, A000203, A000396, A000668, A007947, A057723, A064549, A307958.

f[p_] := 2^p*(2^p-1)^2; f/@MersennePrimeExponent/@Range[10] (* assuming no odd perfect number exists *)

a(n) = A064549(A000396(n)) = A000396(n) * A007947(A000396(n)).

A343443 If n = Product (p_j^k_j) then a(n) = Product (k_j + 2), with a(1) = 1.

Original entry on oeis.org

1, 3, 3, 4, 3, 9, 3, 5, 4, 9, 3, 12, 3, 9, 9, 6, 3, 12, 3, 12, 9, 9, 3, 15, 4, 9, 5, 12, 3, 27, 3, 7, 9, 9, 9, 16, 3, 9, 9, 15, 3, 27, 3, 12, 12, 9, 3, 18, 4, 12, 9, 12, 3, 15, 9, 15, 9, 9, 3, 36, 3, 9, 12, 8, 9, 27, 3, 12, 9, 27, 3, 20, 3, 9, 12, 12, 9, 27, 3, 18

Offset: 1

Ilya Gutkovskiy, Apr 15 2021

Inverse Moebius transform of A056671.

a(n) depends only on the prime signature of n (see formulas). - Bernard Schott, May 03 2021

Cf. A000005, A001221, A005361, A007425, A025847, A034444, A056671, A064549, A074816, A107758, A107759, A348018, A363194.

a[1] = 1; a[n_] := Times @@ ((#[[2]] + 2) & /@ FactorInteger[n]); Table[a[n], {n, 80}]
a[n_] := Sum[If[GCD[d, n/d] == 1, DivisorSigma[0, d], 0], {d, Divisors[n]}]; Table[a[n], {n, 80}]

a(n) = sumdiv(n, d, if(gcd(d, n/d)==1, numdiv(d))) \\ Andrew Howroyd, Apr 15 2021

for(n=1, 100, print1(direuler(p=2, n, (1 + X - X^2)/(1-X)^2)[n], ", ")) \\ Vaclav Kotesovec, Feb 11 2023

from math import prod
from sympy import factorint
def A343443(n): return prod(e+2 for e in factorint(n).values()) # Chai Wah Wu, Feb 21 2025

a(n) = 2^omega(n) * tau_3(n) / tau(n), where omega = A001221, tau = A000005 and tau_3 = A007425.
a(n) = Sum_{d|n, gcd(d, n/d) = 1} tau(d).
From Bernard Schott, May 03 2021: (Start)
a(p^k) = k+2 for p prime, or signature [k].
a(A006881(n)) =  9 for signature [1, 1].
a(A054753(n)) = 12 for signature [2, 1].
a(A065036(n)) = 15 for signature [3, 1].
a(A085986(n)) = 16 for signature [2, 2].
a(A178739(n)) = 18 for signature [4, 1].
a(A143610(n)) = 20 for signature [3, 2].
a(A007304(n)) = 27 for signature [1, 1, 1]. (End)
Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 1/p^s - 1/p^(2*s)). - Vaclav Kotesovec, Feb 11 2023
From Amiram Eldar, Sep 01 2023: (Start)
a(n) = A000005(A064549(n)).
a(n) = A363194(A348018(n)). (End)

A295295 Sum of squarefree divisors of the powerful part of n: a(n) = A048250(A057521(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 6, 1, 4, 3, 1, 1, 1, 3, 1, 1, 1, 12, 1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 3, 8, 6, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 4, 3, 1, 1, 1, 3, 1, 1, 1, 12, 1, 1, 6, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 8, 4, 18, 1, 1, 1, 3, 1

Offset: 1

Antti Karttunen, Nov 25 2017

The sum of the squarefree divisors of n whose square divides n. - Amiram Eldar, Oct 13 2023

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

Cf. A000203, A003557, A048250, A057521, A064549, A092261, A295294, A344137.

Cf. A001620, A013661, A073002.

Array[DivisorSum[#/Denominator[#/Apply[Times, FactorInteger[#][[All, 1]]]^2], # &, SquareFreeQ] &, 105] (* Michael De Vlieger, Nov 26 2017, after Jean-François Alcover at A057521 *)
f[p_, e_] := If[e == 1, 1, p+1] ; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2023 *)

a(n) = my(f=factor(n)); for (i=1, #f~, if (f[i,2]==1, f[i,1]=1)); sumdiv(factorback(f), d, d*issquarefree(d)); \\ Michel Marcus, Jan 29 2021

Multiplicative with a(p) = 1 and a(p^e) = (p+1) for e > 1.
a(n) = A048250(A057521(n)) = A048250(A064549(A003557(n))).
a(n) = A048250(n) / A092261(n).
a(n) = Sum_{d^2|n} d * mu(d)^2. - Wesley Ivan Hurt, Feb 13 2022
From Amiram Eldar, Sep 18 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-1) / zeta(4*s-2).
Sum_{k=1..n} a(k) ~ (3*n/Pi^2) * (log(n) + 3*gamma - 1 - 4*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). (End)
a(n) = A048250(n) - A344137(n). - Amiram Eldar, Oct 13 2023

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

A380987 Position of first appearance of n in A290106 (product of prime indices divided by product of distinct prime indices).

Original entry on oeis.org

1, 9, 25, 27, 121, 169, 289, 81, 125, 841, 961, 675, 1681, 1849, 2209, 243, 3481, 1125, 4489, 3267, 5329, 6241, 6889, 2025, 1331, 10201, 625, 7803, 11881, 12769, 16129, 729, 18769, 19321, 22201, 2197, 24649, 26569, 27889, 9801, 32041, 32761, 36481, 25947

Offset: 1

Gus Wiseman, Feb 14 2025

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

All terms are odd.

			The first position of 12 in A290106 is 675, with prime indices {2,2,2,3,3}, so a(12) = 675.
The terms together with their prime indices begin:
      1: {}
      9: {2,2}
     25: {3,3}
     27: {2,2,2}
    121: {5,5}
    169: {6,6}
    289: {7,7}
     81: {2,2,2,2}
    125: {3,3,3}
    841: {10,10}
    961: {11,11}
    675: {2,2,2,3,3}
   1681: {13,13}
   1849: {14,14}
   2209: {15,15}
    243: {2,2,2,2,2}
   3481: {17,17}
   1125: {2,2,3,3,3}

For factors instead of indices we have A064549 (sorted A001694), firsts of A003557.
The additive version for factors is A280286 (sorted A381075), firsts of A280292.
Position of first appearance of n in A290106.
The additive version is A380956 (sorted A380957), firsts of A380955.
For difference instead of quotient see A380986.
The sorted version is A380988.
A000040 lists the primes, differences A001223.
A003963 gives product of prime indices, distinct A156061.
A005117 lists squarefree numbers, complement A013929.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, length A001222.
A304038 lists distinct prime indices, sum A066328, length A001221.
Cf. A000720, A007947, A046660, A066503, A071625, A136565, A178503, A175508, A325034, A379681.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
q=Table[Times@@prix[n]/Times@@Union[prix[n]],{n,10000}];
Table[Position[q,k][[1,1]],{k,mnrm[q]}]

cp's OEIS Frontend

A280286 a(n) is the least k such that sopfr(k) - sopf(k) = n.

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A356191 a(n) is the smallest exponentially odd number that is divisible by n.

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A327564 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)^(k_j - 1)).

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A073353 Sum of n and its squarefree kernel.

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A295294 Sum of the divisors of the powerful part of n: a(n) = A000203(A057521(n)).

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A307959 Primitive coreful perfect numbers: powerful numbers k such that csigma(k) = 2*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

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A343443 If n = Product (p_j^k_j) then a(n) = Product (k_j + 2), with a(1) = 1.

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