A001157 -id:A001157 - cp's OEIS frontend

A064602 Partial sums of A001157: Sum_{j=1..n} sigma_2(j).

1, 6, 16, 37, 63, 113, 163, 248, 339, 469, 591, 801, 971, 1221, 1481, 1822, 2112, 2567, 2929, 3475, 3975, 4585, 5115, 5965, 6616, 7466, 8286, 9336, 10178, 11478, 12440, 13805, 15025, 16475, 17775, 19686, 21056, 22866, 24566, 26776, 28458, 30958

Offset: 1

Labos Elemer, Sep 24 2001

nonn

In general, for m >= 0 and j >= 0, Sum_{k=1..n} k^m * sigma_j(k) = Sum_{k=1..s} (k^m * F_{m+j}(floor(n/k)) + k^(m+j) * F_m(floor(n/k))) - F_{m+j}(s) * F_m(s), where s = floor(sqrt(n)) and F_m(x) are the Faulhaber polynomials defined as F_m(x) = (Bernoulli(m+1, x+1) - Bernoulli(m+1, 1)) / (m+1). - Daniel Suteu, Nov 27 2020

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Project Euler, Problem 401: Sum of squares of divisors, 2012.

Cf. A001157, A064605.

Cf. A064603, A064604, A248076.

Accumulate@ Array[DivisorSigma[2, #] &, 42] (* Michael De Vlieger, Jan 02 2017 *)

a(n) = sum(j=1, n, sigma(j, 2)); \\ Michel Marcus, Dec 15 2013

f(n) = n*(n+1)*(2*n+1)/6; \\ A000330
a(n) = my(s=sqrtint(n)); sum(k=1, s, f(n\k) + k^2*(n\k)) - s*f(s); \\ Daniel Suteu, Nov 26 2020

from math import isqrt
def f(n): return n*(n+1)*(2*n+1)//6
def a(n):
    s = isqrt(n)
    return sum(f(n//k) + k*k*(n//k) for k in range(1, s+1)) - s*f(s)
print([a(k) for k in range(1, 43)]) # Michael S. Branicky, Oct 01 2022 after Daniel Suteu

a(n) = a(n-1) + A001157(n) = Sum_{j=1..n} sigma_2(j) where sigma_2(j) = A001157(j).
a(n) = Sum_{i=1..n} i^2 * floor(n/i). - Enrique Pérez Herrero, Sep 15 2012
G.f.: (1/(1 - x))*Sum_{k>=1} k^2*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 02 2017
a(n) ~ zeta(3) * n^3 / 3. - Vaclav Kotesovec, Sep 02 2018
a(n) = Sum_{k=1..s} (A000330(floor(n/k)) + k^2*floor(n/k)) - s*A000330(s), where s = floor(sqrt(n)). - Daniel Suteu, Nov 26 2020

A053822 Dirichlet inverse of sigma_2 function (A001157).

Original entry on oeis.org

1, -5, -10, 4, -26, 50, -50, 0, 9, 130, -122, -40, -170, 250, 260, 0, -290, -45, -362, -104, 500, 610, -530, 0, 25, 850, 0, -200, -842, -1300, -962, 0, 1220, 1450, 1300, 36, -1370, 1810, 1700, 0, -1682, -2500, -1850, -488, -234, 2650, -2210, 0, 49, -125, 2900, -680

Offset: 1

N. J. A. Sloane, Apr 08 2000

sigma_2(n) is the sum of the squares of the divisors of n (A001157).

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 39.

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

Dirichlet inverse of sigma_k(n): A007427 (k = 0), A046692 (k = 1), A053825 (k = 3), A053826 (k = 4), A178448 (k = 5).

Cf. A001157,.

f1:= proc(p,e) if e = 1 then -1-p^2 elif e=2 then p^2 else 0 fi end proc:
f:= n -> mul(f1(t[1],t[2]),t=ifactors(n)[2]);
map(f, [$1..100]); # Robert Israel, Jan 29 2018

a[n_] := Sum[MoebiusMu[n/d] MoebiusMu[d] d^2, {d, Divisors[n]}];
Array[a, 100] (* Jean-François Alcover, Mar 05 2019, after Ilya Gutkovskiy *)
f[p_, e_] := If[e == 1, -p^2 - 1, If[e == 2, p^2, 0]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *)

seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, sigma(n, 2)))} \\ Andrew Howroyd, Aug 05 2018

for(n=1, 100, print1(direuler(p=2, n, (1 - X)*(1 - p^2*X))[n], ", ")) \\ Vaclav Kotesovec, Sep 16 2020

Dirichlet g.f.: 1/(zeta(s)*zeta(s-2)).
Multiplicative with a(p^1) = -1-p^2, a(p^2) = p^2, a(p^e) = 0 for e>=3. - Mitch Harris, Jun 27 2005
a(n) = Sum_{d|n} mu(n/d)*mu(d)*d^2. - Ilya Gutkovskiy, Nov 06 2018
From Peter Bala, Jan 26 2024: (Start)
a(n) = Sum_{d divides n} d * (sigma(d))^(-1) * phi(n/d), where (sigma(n))^(-1) = A046692(n) denotes the Dirichlet inverse of sigma(n) = A000203(n).
a(n) = Sum_{d divides n} d^2 * (sigma_k(d))^(-1) * J_(k+2)(n/d) for k >= 0, where (sigma_k(n))^(-1) denotes the Dirichlet inverse of the divisor sum function sigma_k(n) and J_k(n) denotes the Jordan totient function. (End)

A082069 Smallest common prime-divisor of n and Sigma_2(n) = A001157(n); a(n) = 1 if no common prime-divisor exists.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 5, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 5, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 5, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 7, 1, 5, 2, 1, 2, 1, 2, 1, 1, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 5, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 5

Offset: 1

Labos Elemer, Apr 07 2003

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A001157, A020639, A179930.

Cf. also A082063, A082067, A082068, A082070, A082071, A082072.

ffi[x_] := Flatten[FactorInteger[x]]; lf[x_] := Length[FactorInteger[x]]; ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; f1[x_] := n; f2[x_] := DivisorSigma[2, x]; Table[Min[Intersection[ba[f1[w]], ba[f2[w]]]], {w, 1, 128}]
(* Second program: *)
Array[If[CoprimeQ[#1, #2], 1, Min@ Apply[Intersection, Map[FactorInteger[#][[All, 1]] &, {#1, #2}]]] & @@ {#, DivisorSigma[2, #]} &, 105] (* Michael De Vlieger, Nov 03 2017 *)

A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
A082069(n) = A020639(gcd(sigma(n,2), n)); \\ Antti Karttunen, Nov 03 2017

a(n) = A020639(A179930(n)). - Antti Karttunen, Nov 03 2017

Changed "was found" to "exists" in definition. - N. J. A. Sloane, Jan 29 2022

A082065 Greatest common prime-divisor of phi(n)=A000010(n) and sigma(2,n) = A001157(n); a(n) = 1 if no common prime-divisor exists.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 5, 2, 2, 1, 2, 2, 3, 2, 2, 2, 1, 5, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 5, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 2, 1, 2, 5, 2, 2, 2, 2, 2, 1, 2, 2, 5, 3, 5, 2, 2, 2, 1, 5, 2, 3, 2, 2, 2, 5, 2, 2, 2, 2, 5, 2, 2, 2, 2, 3, 2, 1

Offset: 1

Labos Elemer, Apr 07 2003

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A006530, A001157, A000010, A082061-A082066.

gcpd := proc(a,b) local g ,d ; g := 1 ; for d in numtheory[divisors](a) intersect numtheory[divisors](b) do if isprime(d) then g := max(g,d) ; end if; end do: g ; end proc:
A082065 := proc(n) gcpd( numtheory[phi](n), numtheory[sigma][2](n) ) ; end proc:
seq(A082065(n),n=1..120) ; # R. J. Mathar, Jul 09 2011

Table[FactorInteger[GCD[EulerPhi@ n, DivisorSigma[2, n]]][[-1, 1]], {n, 100}] (* Michael De Vlieger, Jul 22 2017 *)

gpf(n)=if(n>1,my(f=factor(n)[,1]);f[#f],1)
a(n)=gpf(gcd(eulerphi(n),sigma(n,2))) \\ Charles R Greathouse IV, Feb 21 2013

Values corrected by R. J. Mathar, Jul 09 2011

Changed "was found" to "exists" in definition. - N. J. A. Sloane, Jan 29 2022

A082066 Greatest common prime-divisor of sigma_1(n)=A000203(n) and sigma_2(n)=A001157(n); a(n)=1 if no common prime-divisor exists.

Original entry on oeis.org

1, 1, 2, 7, 2, 2, 2, 5, 13, 2, 2, 7, 2, 2, 2, 31, 2, 13, 2, 7, 2, 2, 2, 5, 31, 2, 5, 7, 2, 2, 2, 7, 2, 2, 2, 13, 2, 5, 2, 5, 2, 2, 2, 7, 13, 2, 2, 31, 19, 31, 2, 7, 2, 5, 2, 5, 5, 5, 2, 7, 2, 2, 13, 127, 2, 2, 2, 7, 2, 2, 2, 13, 2, 2, 31, 7, 2, 2, 2, 31, 11, 2, 2, 7, 2, 2, 5, 5, 2, 13, 2, 7, 2, 2, 2, 7, 2

Offset: 1

Labos Elemer, Apr 07 2003

nonn

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

Cf. A006530, A001157, A000203, A082061-A082065.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] f1[x_] := DivisorSigma[1, n]; f2[x_] := DivisorSigma[2, x] Table[Max[Intersection[ba[f1[w]], ba[f2[w]]]], {w, 1, 128}]
(* Second program: *)
Table[Last[Apply[Intersection, FactorInteger[Map[DivisorSigma[#, n] &, {1, 2}]][[All, All, 1]]] /. {} -> {1}], {n, 109}] (* Michael De Vlieger, May 22 2017 *)

gpf(n)=if(n>1,my(f=factor(n)[,1]);f[#f],1)
a(n)=gpf(gcd(sigma(n),sigma(n,2))) \\ Charles R Greathouse IV, Feb 19 2013

from sympy import primefactors, gcd, divisor_sigma
def a006530(n): return 1 if n==1 else primefactors(n)[-1]
def a(n): return a006530(gcd(divisor_sigma(n), divisor_sigma(n, 2))) # Indranil Ghosh, May 22 2017

a(n) = A006530(A179931(n)). - Reinhard Zumkeller, Jul 10 2011

Changed "was found" to "exists" in definition. - N. J. A. Sloane, Jan 29 2022

A082063 Greatest common prime divisor of n and sigma_2(n) = A001157(n), or 1 if the two are relatively prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 3, 1, 2, 5, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 7, 1, 5, 1, 1, 1, 2, 5, 3, 1, 2, 1, 5, 1, 2, 1, 2, 1, 2, 1, 2, 1, 5, 1, 2, 1, 2, 1, 2, 1, 2, 1, 5, 1, 2, 7, 1, 13, 2, 1, 2, 1, 5, 1, 1, 1, 2, 5, 2, 1, 2, 1, 2, 1, 2, 1, 7, 5, 2, 1, 2, 1, 5, 1, 2, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 5

Offset: 1

Labos Elemer, Apr 07 2003

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A006530, A001157, A179930.

Cf. also A082061, A082062, A082064, A082065, A082066, A082069.

(* factors/exponent SET *) ffi[x_] := Flatten[FactorInteger[x]]; lf[x_] := Length[FactorInteger[x]]; ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; f1[x_] := x; f2[x_] := DivisorSigma[2, x]; Table[Max[Intersection[ba[f1[w]], ba[f2[w]]]], {w, 1, 128}]
(* Second program: *)
Array[If[CoprimeQ[#1, #2], 1, Max@ Apply[Intersection, Map[FactorInteger[#][[All, 1]] &, {#1, #2}]]] & @@ {#, DivisorSigma[2, #]} &, 105] (* Michael De Vlieger, Nov 03 2017 *)

A006530(n) = if(1==n, n, vecmax(factor(n)[, 1]));
A082063(n) = A006530(gcd(sigma(n,2), n)); \\ Antti Karttunen, Nov 03 2017

a(n) = A006530(A179930(n)). - Antti Karttunen, Nov 03 2017

Erroneous comment removed by Antti Karttunen, Nov 03 2017

A082071 Smallest common prime-divisor of phi(n) = A000010(n) and sigma_2(n) = A001157(n); a(n)=1 if no common prime-divisor exists.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 1, 2, 2, 2, 2, 2

Offset: 1

Labos Elemer, Apr 07 2003

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A001157, A020639.

Cf. also A082065, A082067, A082068, A082069, A082070, A082072.

Array[If[CoprimeQ[#1, #2], 1, Min@ Apply[Intersection, Map[FactorInteger[#][[All, 1]] &, {#1, #2}]]] & @@ {EulerPhi@ #,
DivisorSigma[2, #]} &, 105] (* Michael De Vlieger, Nov 03 2017 *)

A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));
A082071(n) = A020639(gcd(eulerphi(n),sigma(n,2))); \\ Antti Karttunen, Nov 03 2017

a(n) = A020639(gcd(A000010(n), A001157(n))). - Antti Karttunen, Nov 03 2017

Values corrected by R. J. Mathar, Jul 09 2011
More terms from Antti Karttunen, Nov 03 2017
Changed "was found" to "exists" in definition. - N. J. A. Sloane, Jan 29 2022

A082072 Smallest prime that divides sigma(n) = A000203(n) and sigma_2(n) = A001157(n), or 1 if sigma(n) and sigma_2(n) are relatively prime.

Original entry on oeis.org

1, 1, 2, 7, 2, 2, 2, 5, 13, 2, 2, 2, 2, 2, 2, 31, 2, 13, 2, 2, 2, 2, 2, 2, 31, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 7, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 127, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 11, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 7, 2, 2

Offset: 1

Labos Elemer, Apr 07 2003

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000203, A001157, A020639, A179931.

Cf. also A082066, A082067, A082068, A082069, A082070, A082071.

ffi[x_] := Flatten[FactorInteger[x]]; lf[x_] := Length[FactorInteger[x]]; ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; f1[x_] := DivisorSigma[1, x]; f2[x_] := DivisorSigma[2, x]; Table[Min[Intersection[ba[f1[w]], ba[f2[w]]]], {w, 1, 128}]
(* Second program: *)
Array[If[CoprimeQ[#1, #2], 1, Min@ Apply[Intersection, Map[FactorInteger[#][[All, 1]] &, {#1, #2}]]] & @@ {DivisorSigma[1, #], DivisorSigma[2, #]} &, 105] (* Michael De Vlieger, Nov 03 2017 *)

lpf(n)=my(f=factor(n)[,1]); if(#f,f[1],1)
a(n)=lpf(gcd(sigma(n),sigma(n,2))) \\ Charles R Greathouse IV, Feb 14 2013

a(n) = A020639(A179931(n)). - Antti Karttunen, Nov 03 2017

Name edited by Antti Karttunen after an example by N. J. A. Sloane, Nov 04 2017

A203849 a(n) = sigma_2(n)*Fibonacci(n), where sigma_2(n) = A001157(n), the sum of squares of divisors of n.

Original entry on oeis.org

1, 5, 20, 63, 130, 400, 650, 1785, 3094, 7150, 10858, 30240, 39610, 94250, 158600, 336567, 463130, 1175720, 1513522, 3693690, 5473000, 10803710, 15188210, 39412800, 48841275, 103184050, 161062760, 333701550, 432980818, 1081652000, 1295110778, 2973391785, 4299985160

Offset: 1

Paul D. Hanna, Jan 12 2012

nonn

Compare g.f. to the Lambert series identity: Sum_{n>=1} n^2*x^n/(1-x^n) = Sum_{n>=1} sigma_2(n)*x^n.

			G.f.: A(x) = x + 5*x^2 + 20*x^3 + 63*x^4 + 130*x^5 + 400*x^6 + 650*x^7 +...
where A(x) = x/(1-x-x^2) + 2^2*1*x^2/(1-3*x^2+x^4) + 3^2*2*x^3/(1-4*x^3-x^6) + 4^2*3*x^4/(1-7*x^4+x^8) + 5^2*5*x^5/(1-11*x^5-x^10) + 6^2*8*x^6/(1-18*x^6+x^12) +...+ n^2*fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...

G. C. Greubel, Table of n, a(n) for n = 1..2500

Cf. A203847, A203848, A203838, A001157 (sigma_2), A000204 (Lucas), A000045.

Table[DivisorSigma[2, n]*Fibonacci[n], {n, 50}] (* G. C. Greubel, Jul 17 2018 *)

PARI
```
{a(n)=sigma(n,2)*fibonacci(n)}
```

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(sum(m=1,n,m^2*fibonacci(m)*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}

G.f.: Sum_{n>=1} n^2*fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} sigma_2(n)*fibonacci(n)*x^n, where Lucas(n) = A000204(n).

A204272 a(n) = sigma_2(n)*Pell(n), where sigma_2(n) = A001157(n), the sum of squares of divisors of n.

Original entry on oeis.org

1, 10, 50, 252, 754, 3500, 8450, 34680, 89635, 309140, 700402, 2910600, 5688370, 20195500, 50706500, 160553712, 329639810, 1248615550, 2398289458, 8732957688, 19306982500, 56865638380, 119281100930, 461838762000, 853941516771

Offset: 1

Paul D. Hanna, Jan 14 2012

nonn

Compare g.f. to the Lambert series identity: Sum_{n>=1} n^2*x^n/(1-x^n) = Sum_{n>=1} sigma_2(n)*x^n.

			G.f.: A(x) = x + 10*x^2 + 50*x^3 + 252*x^4 + 754*x^5 + 3500*x^6 +...
where A(x) = x/(1-2*x-x^2) + 2^2*2*x^2/(1-6*x^2+x^4) + 3^2*5*x^3/(1-14*x^3-x^6) + 4^2*12*x^4/(1-34*x^4+x^8) + 5^2*29*x^5/(1-82*x^5-x^10) + 6^2*70*x^6/(1-198*x^6+x^12) +...+ n^2*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) +...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A203849, A204270, A204271, A204273, A204274, A204275, A001157 (sigma_2), A002203, A000045.

With[{nn=30},Times@@@Thread[{Rest[LinearRecurrence[{2,1},{0,1},nn+1]], DivisorSigma[ 2,Range[nn]]}]] (* Harvey P. Dale, Oct 21 2015 *)

/* Subroutines used in PARI programs below: */
{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)), n)}
{A002203(n)=polcoeff(2*(1-x)/(1-2*x-x^2+x*O(x^n)), n)}

PARI
```
{a(n)=sigma(n,2)*Pell(n)}
```

{a(n)=polcoeff(sum(m=1,n,m^2*Pell(m)*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}

G.f.: Sum_{n>=1} n^2*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} sigma_2(n)*Pell(n)*x^n, where Pell(n) = A000129(n) and A002203 is the companion Pell numbers.

cp's OEIS Frontend

A064602 Partial sums of A001157: Sum_{j=1..n} sigma_2(j).

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A053822 Dirichlet inverse of sigma_2 function (A001157).

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A082069 Smallest common prime-divisor of n and Sigma_2(n) = A001157(n); a(n) = 1 if no common prime-divisor exists.

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A082065 Greatest common prime-divisor of phi(n)=A000010(n) and sigma(2,n) = A001157(n); a(n) = 1 if no common prime-divisor exists.

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A082066 Greatest common prime-divisor of sigma_1(n)=A000203(n) and sigma_2(n)=A001157(n); a(n)=1 if no common prime-divisor exists.

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A082063 Greatest common prime divisor of n and sigma_2(n) = A001157(n), or 1 if the two are relatively prime.

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A082071 Smallest common prime-divisor of phi(n) = A000010(n) and sigma_2(n) = A001157(n); a(n)=1 if no common prime-divisor exists.

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