A009223 -id:A009223 - 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}]

A222713 Least number k such that n divides gcd(sigma(k), phi(k)) (A009223).

Original entry on oeis.org

1, 3, 14, 12, 88, 14, 116, 15, 190, 88, 989, 35, 477, 116, 209, 105, 6901, 190, 7067, 88, 196, 989, 6439, 35, 15049, 477, 2754, 172, 10207, 209, 4976, 336, 989, 6901, 1189, 190, 10877, 7067, 477, 248, 13529, 377, 44461, 989, 418, 6439, 79523, 105, 10244, 15049

Offset: 1

Phil Carmody, Mar 01 2013

nonn

For each n there are infinitely many numbers k for which n divides sigma(k) and phi(k). - Marius A. Burtea, Mar 28 2019

			Given A009223 = 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 4, 2, 6, 8, 1, 2, 3, ...
1 first divides A009223(1); 2 first divides A009223(3); 3 first divides A009223(14)=6.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (terms 1-388 from Marius A. Burtea, 389-2808 from David A. Corneth)

Cf. A009223, A222714.

[Min([n: n in [1..300000] | IsIntegral(SumOfDivisors(n)/m) and IsIntegral(EulerPhi(n)/m) ]): m in [1..70]]; // Marius A. Burtea, Mar 28 2019

v:=[];
for n in [1..60] do
m:=1;
        while  not EulerPhi(m) mod n  eq 0 or not SumOfDivisors(m) mod n  eq 0 do
           v[n]:=0;
           m:=m+1;
        end while;
     v[n]:=m;
end for;
v; // Marius A. Burtea, Mar 30 2019

Array[Block[{i = 1}, While[Mod[GCD[DivisorSigma[1, i], EulerPhi@ i], #] != 0, i++]; i] &, 50] (* Michael De Vlieger, Mar 28 2019 *)

a(n)={my(k=1); while(gcd(sigma(k), eulerphi(k))%n!=0, k++); k}

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

A222711 Numbers k such that gcd(sigma(k), phi(k)) (A009223) attains record values.

Original entry on oeis.org

1, 3, 12, 14, 15, 35, 105, 190, 248, 357, 616, 812, 1045, 3080, 3135, 3339, 4064, 5049, 8323, 8636, 10659, 12441, 16065, 19780, 20026, 23374, 24871, 29029, 50065, 58435, 64285, 87685, 124355, 132957, 137885, 140335, 248501, 263055, 317205, 353133, 423657, 596037, 655707, 734517, 894387

Offset: 1

Phil Carmody, Mar 01 2013

RECORDS transform of A009223.

Donovan Johnson, Table of n, a(n) for n = 1..500
N. J. A. Sloane, Transforms (The RECORDS transform returns both the high-water marks and the places where they occur).

Cf. A009223, A222712.

a[1] = 1; record = 1; a[n_] := a[n] = For[k = a[n-1] + 1, True, k++, g = GCD[DivisorSigma[1, k], EulerPhi[k]]; If[g > record, record = g; Return[k]]]; Table[a[n], {n, 1, 45}] (* Jean-François Alcover, Oct 07 2013 *)
DeleteDuplicates[Table[{k,GCD[DivisorSigma[1,k],EulerPhi[k]]},{k,900000}],GreaterEqual[ #1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Jun 22 2024 *)

mg=0;for(x=1,1000000,g=A009223(x);if(g>mg,print1(x", ");mg=g))

A222712 Record values of gcd(sigma(n), phi(n)) (A009223).

Original entry on oeis.org

1, 2, 4, 6, 8, 24, 48, 72, 120, 192, 240, 336, 720, 960, 1440, 1872, 2016, 2880, 3360, 4032, 5760, 6720, 6912, 7392, 8640, 10080, 17280, 20160, 34560, 40320, 44352, 60480, 69120, 74880, 95040, 96768, 100800, 120960, 134784, 201600, 241920, 322560, 354816, 411840, 483840

Offset: 1

Phil Carmody, Mar 01 2013

nonn

RECORDS transform of A009223.

Donovan Johnson, Table of n, a(n) for n = 1..500
N. J. A. Sloane, Transforms (The RECORDS transform returns both the high-water marks and the places where they occur).

Cf. A009223, A222711 (indexes where these records are attained).

mg=0;for(x=1,1000000,g=A009223(x);if(g>mg,print1(g", ");mg=g))

A222714 Smallest i such that prime(n) divides gcd(sigma(i), phi(i)) (cf. A009223).

Original entry on oeis.org

3, 14, 88, 116, 989, 477, 6901, 7067, 6439, 10207, 4976, 10877, 13529, 44461, 79523, 22577, 250277, 62023, 107869, 161027, 75008, 49769, 55277, 183296, 75077, 612463, 381923, 412163, 712423, 153679, 32576, 137549, 450181, 154289, 1776377, 1642577, 491723, 637981, 3903791, 239777, 642251, 1572889, 1608983, 1192739, 2791489, 316409, 888731, 4773091, 4942243, 1256293

Offset: 1

Phil Carmody, Mar 01 2013

nonn

			Given A009223 = 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 4, 2, 6, 8, 1, 2, 3, ...
prime(1)=2 first divides A009223(3); prime(2)=3 first divides A009223(14)=6; prime(3)=5 first divides both sigma(88)=180 and phi(88)=40, so A222714(3)=88.

Cf. A009223. Subsequence of A222713.

A009223_hunt(x)=local(n=0,g);while(n++,g=A009223(n);if(g%x,,return(n)));
for(x=1,50,print1(A009223_hunt(prime(x))", "))

A372569 Lexicographically earliest infinite sequence such that a(i) = a(j) => A009194(i) = A009194(j), A009195(i) = A009195(j) and A009223(i) = A009223(j), for all i, j >= 1.

Original entry on oeis.org

1, 1, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 7, 2, 15, 16, 9, 17, 18, 2, 19, 2, 20, 21, 7, 22, 23, 2, 9, 24, 25, 2, 26, 2, 8, 27, 7, 2, 28, 29, 30, 10, 13, 2, 31, 32, 33, 14, 7, 2, 34, 2, 9, 35, 36, 37, 38, 2, 13, 21, 39, 2, 40, 2, 9, 41, 8, 37, 42, 2, 43, 44, 7, 2, 45, 46, 9, 10, 47, 2, 48, 49, 8, 14, 7, 50, 51, 2, 52, 53, 54, 2, 19, 2

Offset: 1

Antti Karttunen, May 25 2024

nonn

Restricted growth sequence transform of the triple [A009194(n), A009195(n), A009223(n)].

For all i, j: A372570(i) = A372570(j) => a(i) = a(j) => A074389(i) = A074389(j).

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

Cf. A009194, A009195, A009223.

Cf. also A372570, A372572.

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; };
Aux372569(n) = [gcd(n, sigma(n)), gcd(n, eulerphi(n)), gcd(eulerphi(n), sigma(n))];
v372569 = rgs_transform(vector(up_to, n, Aux372569(n)));
A372569(n) = v372569[n];

A372570 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A009194(n), A009195(n), A009223(n), A322361(n), A342671(n)], for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 7, 8, 3, 9, 3, 10, 11, 12, 3, 13, 3, 14, 15, 8, 3, 16, 17, 10, 18, 19, 3, 20, 3, 21, 22, 8, 23, 24, 3, 10, 25, 26, 3, 27, 3, 28, 29, 8, 3, 30, 31, 32, 33, 34, 3, 35, 36, 37, 38, 8, 3, 39, 3, 10, 40, 41, 42, 43, 3, 44, 22, 45, 3, 46, 3, 10, 47, 48, 49, 50, 3, 51, 52, 8, 3, 53, 54, 10, 55, 56, 3, 57, 58, 28, 15, 8, 59, 60, 3, 61, 62, 63, 3

Offset: 1

Antti Karttunen, May 25 2024

nonn

Restricted growth sequence transform of the quintuple [A009194(n), A009195(n), A009223(n), A322361(n), A342671(n)].
For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A372569(i) = A372569(j),
  a(i) = a(j) => A372572(i) = A372572(j).

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

Cf. A009194, A009195, A009223, A322361, A342671.

Cf. also A305801, A372569, A372572.

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; };
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
Aux372570(n) = [gcd(n, sigma(n)), gcd(n, eulerphi(n)), gcd(eulerphi(n), sigma(n)), gcd(n, A003961(n)), gcd(sigma(n), A003961(n))];
v372570 = rgs_transform(vector(up_to, n, Aux372570(n)));
A372570(n) = v372570[n];

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}]

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.

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A222713 Least number k such that n divides gcd(sigma(k), phi(k)) (A009223).

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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.

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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.

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A222711 Numbers k such that gcd(sigma(k), phi(k)) (A009223) attains record values.

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A222712 Record values of gcd(sigma(n), phi(n)) (A009223).

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A222714 Smallest i such that prime(n) divides gcd(sigma(i), phi(i)) (cf. A009223).

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A372569 Lexicographically earliest infinite sequence such that a(i) = a(j) => A009194(i) = A009194(j), A009195(i) = A009195(j) and A009223(i) = A009223(j), for all i, j >= 1.

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A372570 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A009194(n), A009195(n), A009223(n), A322361(n), A342671(n)], for all i, j >= 1.