A066086 -id:A066086 - cp's OEIS frontend

A076485 Solutions to gcd(sigma(x), phi(x)) > gcd(sigma(core(x)), phi(core(x))), i.e., when A009223(x) > A066086(x) or if A066087(x) > 0.

12, 18, 24, 44, 48, 49, 54, 56, 72, 88, 92, 96, 99, 108, 112, 116, 125, 132, 135, 140, 147, 152, 162, 168, 169, 172, 176, 184, 188, 192, 196, 198, 200, 207, 216, 224, 236, 248, 250, 264, 270, 276, 280, 284, 288, 297, 308, 328, 332, 336, 344, 348, 352, 361

Offset: 1

Labos Elemer, Oct 17 2002

nonn

			For n=12: sigma(12)=28, phi(12)=4, gcd(28,4)=4 core(12)=6, sigma(6)=12, phi(6)=2, gcd(12,2)=2.

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

Cf. A000010, A000203, A007947, A009223, A023900, A048250, A066086, A066087.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] cor[x_] := Apply[Times, ba[x]] g1[x_] := GCD[DivisorSigma[1, x], EulerPhi[x]] g2[x_] := GCD[DivisorSigma[1, cor[x]], EulerPhi[cor[x]]] Do[s1=g1[n]; s2=g2[n]; If[Greater[s1, s2], Print[n]], {n, 1, 256}]

A076486 Solutions to gcd(sigma(x), phi(x)) < gcd(sigma(core(x)), phi(core(x))), i.e., when A009223(x) < A066086(x) or if A066087(x) < 0.

Original entry on oeis.org

9, 25, 28, 36, 45, 50, 52, 75, 76, 81, 84, 90, 98, 100, 117, 121, 124, 144, 148, 150, 153, 156, 175, 180, 208, 225, 228, 234, 242, 244, 245, 252, 261, 268, 275, 289, 292, 300, 304, 306, 316, 324, 325, 333, 338, 360, 364, 369, 372, 380, 388, 392, 400, 405, 412

Offset: 1

Labos Elemer, Oct 17 2002

nonn

			For n=9: sigma(9)=13, phi(9)=6, gcd(13,6)=1, core(9)=3, sigma(3)=4, phi(3)=2, gcd(4,2)=2.

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

Cf. A000010, A000203, A007947, A009223, A023900, A048250, A066086, A066087, A076485.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] cor[x_] := Apply[Times, ba[x]] g1[x_] := GCD[DivisorSigma[1, x], EulerPhi[x]] g2[x_] := GCD[DivisorSigma[1, cor[x]], EulerPhi[cor[x]]] Do[s1=g1[n]; s2=g2[n]; If[Greater[s2, s1], Print[n]], {n, 1, 256}]

A076487 Solutions to gcd(sigma(x), phi(x)) = gcd(sigma(core(x)), phi(core(x))), i.e., when A009223(x) = A066086(x) or if A066087(x) = 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 77, 78, 79, 80, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95

Offset: 1

Labos Elemer, Oct 17 2002

nonn

The squarefree numbers are a subset of this sequence.

			For n=20: sigma(20)=42, phi(20)=8, gcd(42,8)=2, core(20)=10, sigma(10)=18, phi(10)=4, gcd(18,4)=2, so A009223(20) = A066086(20)=2.

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

Cf. A000010, A000203, A007947, A009223, A023900, A048250, A066086, A066087, A076485, A076486.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] cor[x_] := Apply[Times, ba[x]] g1[x_] := GCD[DivisorSigma[1, x], EulerPhi[x]] g2[x_] := GCD[DivisorSigma[1, cor[x]], EulerPhi[cor[x]]] Do[s1=g1[n]; s2=g2[n]; If[Equal[s2, s1], Print[n]], {n, 1, 256}]

isok(n) = my(c=core(n)); gcd(sigma(n), eulerphi(n)) == gcd(sigma(c), eulerphi(c)); \\ Michel Marcus, Jul 30 2017

A076488 Nonsquarefree solutions to gcd(sigma(x), phi(x)) = gcd(sigma(core(x)), Phi(core(x))), i.e., when A009223(x) = A066086(x) or if A066087(x)=0 and mu(x)=0.

Original entry on oeis.org

4, 8, 16, 20, 27, 32, 40, 60, 63, 64, 68, 80, 104, 120, 126, 128, 136, 160, 164, 171, 189, 204, 212, 220, 232, 240, 243, 256, 260, 272, 279, 294, 296, 312, 315, 320, 340, 342, 343, 350, 351, 356, 363, 375, 378, 387, 404, 408, 416, 424, 464, 476, 480, 492, 512

Offset: 1

Labos Elemer, Oct 17 2002

nonn

			n=60: sigma(60)=168, phi(60)=16, gcd(168,16)=8, core(60)=30, sigma(30)=72, phi(30)=8, gcd(72,8)=8, so A009223(60)=A066086(60)=8.

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

Cf. A066086, A066087, A048250, A023900, A000203, A007947, A000010, A009223, A076485-A076487.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] cor[x_] := Apply[Times, ba[x]] g1[x_] := GCD[DivisorSigma[1, x], EulerPhi[x]] g2[x_] := GCD[DivisorSigma[1, cor[x]], EulerPhi[cor[x]]] Do[s1=g1[n]; s2=g2[n]; If[Equal[s2, s1]&&Equal[MoebiusMu[n], 0], Print[n]], {n, 1, 1024}]

A322591 Lexicographically earliest such sequence a that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = 0 for odd primes, and A007947(n) for any other number.

Original entry on oeis.org

1, 2, 3, 2, 3, 4, 3, 2, 5, 6, 3, 4, 3, 7, 8, 2, 3, 4, 3, 6, 9, 10, 3, 4, 11, 12, 5, 7, 3, 13, 3, 2, 14, 15, 16, 4, 3, 17, 18, 6, 3, 19, 3, 10, 8, 20, 3, 4, 21, 6, 22, 12, 3, 4, 23, 7, 24, 25, 3, 13, 3, 26, 9, 2, 27, 28, 3, 15, 29, 30, 3, 4, 3, 31, 8, 17, 32, 33, 3, 6, 5, 34, 3, 19, 35, 36, 37, 10, 3, 13, 38, 20, 39, 40, 41, 4, 3, 7, 14, 6, 3, 42, 3, 12, 43

Offset: 1

Antti Karttunen, Dec 18 2018

nonn

For all i, j:
  a(i) = a(j) => A066086(i) = A066086(j),
  a(i) = a(j) => A322354(i) = A322354(j).

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

Cf. A007947, A305801, A322587, A322588, A322589, A322590, A322592.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A007947(n) = factorback(factorint(n)[, 1]);
Aux322591(n) = if((n>2)&&isprime(n),0,A007947(n));
v322591 = rgs_transform(vector(up_to, n, Aux322591(n)));
A322591(n) = v322591[n];

A322320 a(n) = gcd(A003557(n), A173557(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1

Offset: 1

Antti Karttunen, Dec 05 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000010, A003557, A173557, A322321, A322351, A322352.

Cf. also A066086, A322318.

a[n_] := If[n == 1, 1, Module[{f=FactorInteger[n]}, GCD[ Times@@ (First[#] ^(Last[#]-1)& /@  f), Times@@((#-1)& @@@ f)]]]; Array[a, 120] (* Amiram Eldar, Dec 05 2018 *)

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
A322320(n) = gcd(A173557(n), A003557(n));

a(n) = gcd(A003557(n), A173557(n)) = gcd(A322351(n), A322352(n)).

a(n) = A000010(n) / A322321(n).

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

A322362 a(n) = gcd(n, A166590(n)), where A166590 is completely multiplicative with a(p) = p+2 for prime p.

Original entry on oeis.org

1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 5, 16, 1, 2, 1, 4, 3, 2, 1, 8, 1, 2, 1, 4, 1, 10, 1, 32, 1, 2, 7, 4, 1, 2, 3, 8, 1, 6, 1, 4, 5, 2, 1, 16, 1, 2, 1, 4, 1, 2, 1, 8, 3, 2, 1, 20, 1, 2, 9, 64, 5, 2, 1, 4, 1, 14, 1, 8, 1, 2, 5, 4, 1, 6, 1, 16, 1, 2, 1, 12, 1, 2, 1, 8, 1, 10, 1, 4, 3, 2, 1, 32, 1, 2, 1, 4, 1, 2, 1, 8, 105

Offset: 1

Antti Karttunen, Dec 05 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A001359, A006512, A037074, A066086, A166590, A322361.

a[n_] := If[n == 1, 1, GCD[n, Times@@ ((First[#]+2)^Last[#] &/@FactorInteger[n])]]; Array[a, 120] (* Amiram Eldar, Dec 05 2018~ *)

A166590(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] += 2); factorback(f); };
A322362(n) = gcd(n, A166590(n));

a(n) = gcd(n, A166590(n)).

a(A037074(n)) = A006512(n).

A066087 a(n) = gcd(sigma(n), phi(n)) - gcd(sigma(rad(n)), phi(rad(n))); rad = A007947.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, -1, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 2, -2, 0, 0, 2, 1, -1, 0, -4, 0, 4, 0, 18, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -4, -2, 0, 0, 0, 0, -1, 0, 0, -4, 0, 0, 0, 18, 0, -2, 0, 2, 0, 0, 0, 2, 0, -3, 8, -1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 12, -6

Offset: 1

Labos Elemer, Dec 04 2001

sign

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

Cf. A048250, A023900, A000203, A007947, A000010, A009223, A066086.

Table[GCD[DivisorSigma[1, n], EulerPhi@ n] - GCD[DivisorSigma[1, #], EulerPhi@ #] &[Times @@ FactorInteger[n][[All, 1]]], {n, 120}] (* Michael De Vlieger, Feb 19 2017 *)

rad(f)=for(i=1,#f~,f[i,2]=1); f
g(f)=gcd(sigma(f),eulerphi(f))
a(n)=my(f=factor(n),k=rad(f)); g(f)-g(k) \\ Charles R Greathouse IV, Dec 09 2013

A009223(n) - A066086(n) = gcd(sigma(n), phi(n)) - gcd(sigma(A007947(n)), phi(A007947(n))).

A322318 a(n) = gcd(A003557(n), A048250(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 4, 1, 1, 1, 2, 1, 3, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 3, 2, 1, 1, 1, 2, 1

Offset: 1

Antti Karttunen, Dec 05 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A001615, A003557, A048250, A322319.

Cf. also A066086, A322320.

a[n_] := If[n == 1, 1, Module[{f=FactorInteger[n]}, GCD[ Times@@ (First[#] ^(Last[#]-1)& /@  f), Times@@((#+1)& @@@ f)]]]; Array[a, 120] (* Amiram Eldar, Dec 05 2018 *)

A003557(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A048250(n) = factorback(apply(p -> p+1, factor(n)[, 1]));
A322318(n) = gcd(A048250(n), A003557(n));

a(n) = gcd(A003557(n), A048250(n)).

a(n) = A001615(n) / A322319(n).

cp's OEIS Frontend

A076485 Solutions to gcd(sigma(x), phi(x)) > gcd(sigma(core(x)), phi(core(x))), i.e., when A009223(x) > A066086(x) or if A066087(x) > 0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A076486 Solutions to gcd(sigma(x), phi(x)) < gcd(sigma(core(x)), phi(core(x))), i.e., when A009223(x) < A066086(x) or if A066087(x) < 0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A076487 Solutions to gcd(sigma(x), phi(x)) = gcd(sigma(core(x)), phi(core(x))), i.e., when A009223(x) = A066086(x) or if A066087(x) = 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A076488 Nonsquarefree solutions to gcd(sigma(x), phi(x)) = gcd(sigma(core(x)), Phi(core(x))), i.e., when A009223(x) = A066086(x) or if A066087(x)=0 and mu(x)=0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A322591 Lexicographically earliest such sequence a that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = 0 for odd primes, and A007947(n) for any other number.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A322320 a(n) = gcd(A003557(n), A173557(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A322362 a(n) = gcd(n, A166590(n)), where A166590 is completely multiplicative with a(p) = p+2 for prime p.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI