A082054 -id:A082054 - cp's OEIS frontend

A082055 Product of common prime-divisors (without multiplicity) of sigma(n) and phi(n).

1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 6, 2, 1, 2, 3, 2, 2, 2, 2, 2, 2, 1, 6, 2, 2, 2, 2, 2, 1, 2, 2, 6, 1, 2, 6, 2, 2, 2, 6, 2, 2, 6, 2, 2, 2, 3, 1, 2, 2, 2, 6, 2, 6, 2, 2, 2, 2, 2, 6, 2, 1, 6, 2, 2, 2, 2, 6, 2, 3, 2, 6, 2, 2, 6, 6, 2, 2, 1, 2, 2, 2, 2, 6, 2, 10, 2, 6, 2, 2, 2, 2, 6, 2, 2, 3, 6, 1, 2, 2, 2, 6, 6

Offset: 1

Labos Elemer, Apr 03 2003

nonn

The squarefree kernel of the greatest common divisor of sigma(n) and phi(n). - Antti Karttunen, Jan 22 2020

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000203, A000010, A007947, A009223, A081396, A082054.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] Table[Apply[Times, Intersection[ba[EulerPhi[w]], ba[DivisorSigma[1, w]]]], {w, 1, 256}]

A082055(n) = factorback(factorint(gcd(sigma(n), eulerphi(n)))[, 1]); \\ Antti Karttunen, Jan 22 2020

a(n) = A007947(A009223(n)). - Antti Karttunen, Jan 22 2020

A082056 Least x = a(n) such that sum of common prime divisors (without multiplicity) of sigma(x) and phi(x) equals n, or 0 if such number (apparently) does not exist.

Original entry on oeis.org

0, 3, 18, 0, 14, 0, 88, 1800, 116, 196, 9801, 377, 2881, 1189, 711, 989, 3596, 477, 6901, 5203, 8473, 9179, 3956, 7067, 6439, 27709, 41309, 10763, 27117, 20569, 10207, 69091, 4976, 15376, 114953, 18650, 204469, 37225, 16279, 130300, 74450, 10877

Offset: 1

Labos Elemer, Apr 03 2003

nonn

A solution is not possible for a(1), a(4) and a(6). - Donovan Johnson, Feb 28 2013

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A000203, A000010, A082054, A082055.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] t=Table[0, {100}]; Do[s=Apply[Plus, Intersection [ba[EulerPhi[n]], ba[DivisorSigma[1, n]]]]; If[s<101&&t[[s]]\[Equal]0, t[[s]]=n], {n, 2, 1000000}]; t

A082057 Least x=a(n) such that product of common prime-divisors [without multiplicity] of sigma(x) and phi(x) equals n; or 0 if n is not a squarefree number or if no such x exists. Among indices n only squarefree numbers arise because multiplicity of prime factors is ignored.

Original entry on oeis.org

1, 3, 18, 0, 200, 14, 3364, 0, 0, 88, 9801, 0, 25281, 116, 1800, 0, 36992, 0, 4414201, 0, 196, 2881, 541696, 0, 0, 711, 0, 0, 98942809, 209, 1547536, 0, 19602, 6901, 814088, 0, 49042009, 8473, 1521, 0, 3150464641, 377, 245178368, 0, 0, 6439, 9265217536, 0, 0

Offset: 1

Labos Elemer, Apr 03 2003

nonn

			For n = 85: a(85) = 924800 = 128*5*5*17*17; sigma(924800) = 2426835 = 3*5*17*31*307; phi(924800) = 348160 = 4096*5*17; common prime factor 5.17 = 85.

Cf. A000203, A000010, A082054, A082055, A082056.

Cf. A073815.

ffi[x_] := Flatten[FactorInteger[x]]
lf[x_] := Length[FactorInteger[x]]
ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]
t=Table[0, {100}]; Do[s=Apply[Times, Intersection
[ba[EulerPhi[n]], ba[DivisorSigma[1, n]]]];
If[s<101&&t[[s]]==0, t[[s]]=n], {n, 1, 1000000}]; t

a(n) = Min{x; A082055(x)=n}; 0 if n is not squarefree.

Corrected and extended by David Wasserman, Aug 27 2004

cp's OEIS Frontend

A082055 Product of common prime-divisors (without multiplicity) of sigma(n) and phi(n).

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A082056 Least x = a(n) such that sum of common prime divisors (without multiplicity) of sigma(x) and phi(x) equals n, or 0 if such number (apparently) does not exist.

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A082057 Least x=a(n) such that product of common prime-divisors [without multiplicity] of sigma(x) and phi(x) equals n; or 0 if n is not a squarefree number or if no such x exists. Among indices n only squarefree numbers arise because multiplicity of prime factors is ignored.

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