A009195 -id:A009195 - cp's OEIS frontend

A064448 a(n) = gcd(n^n, phi(n^n)).

1, 2, 9, 128, 625, 15552, 117649, 8388608, 129140163, 2000000000, 25937424601, 2972033482752, 23298085122481, 1587429546508288, 29192926025390625, 9223372036854775808, 48661191875666868481, 13115469358432179191808

Offset: 1

Labos Elemer, Oct 02 2001

nonn

Harry J. Smith, Table of n, a(n) for n=1..100

Cf. A000010, A000312, A009195.

f:= proc(n) local F,x;
  F:= ifactors(n)[2];
  mul(x[1]^(n*x[2]-1),x=F) * igcd(mul(x[1],x=F), mul(x[1]-1,x=F))
end proc:
map(f, [$1..100]); # Robert Israel, Jan 18 2018

a(n) = { my(p=n^n); gcd(p, eulerphi(p)) } \\ Harry J. Smith, Sep 14 2009

a(n) = gcd(A000312(n), A000010(A000312(n))).
If n = Product_j (p_j)^(e_j) is the prime factorization of n, then a(n) = Product_j p_j^(n e_j - 1) * gcd(Product_j p_j, Product_j (p_j-1)). - Robert Israel, Jan 18 2018
a(n) = A009195(A000312(n)). - Andrew Howroyd, Dec 14 2024

A068502 Composite numbers k such that gcd(sigma(k), k) = gcd(k, phi(k)).

Original entry on oeis.org

10, 12, 14, 22, 26, 34, 35, 38, 42, 44, 45, 46, 56, 58, 62, 65, 70, 74, 76, 77, 78, 82, 85, 86, 92, 94, 99, 105, 106, 114, 115, 118, 119, 122, 124, 130, 133, 134, 142, 143, 146, 154, 158, 161, 166, 168, 170, 172, 178, 184, 185, 186, 187, 188, 194, 195, 202, 206

Offset: 1

Benoit Cloitre, Mar 11 2002

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

Cf. A000010, A000203, A009194, A009195.

Cases[Range[2, 206], n_ /; !PrimeQ[n] && GCD[Total[Divisors[n]], n] == GCD[n, EulerPhi[n]]] (* Jean-François Alcover, Mar 15 2011 *)
fQ[n_]:=!PrimeQ[n]&&GCD[Total[Divisors[n]],n] == GCD[n,EulerPhi[n]];Select[Range[2,206],fQ] (* Zak Seidov, Mar 15 2011 *)

A069809 Numbers k such that gcd(k, phi(k)) = tau(k).

Original entry on oeis.org

1, 8, 9, 18, 24, 40, 56, 84, 88, 104, 136, 152, 156, 184, 228, 232, 248, 296, 328, 344, 360, 372, 376, 424, 444, 472, 488, 516, 536, 568, 584, 632, 664, 712, 732, 776, 792, 804, 808, 824, 856, 872, 876, 904, 948, 1016, 1048, 1096, 1112, 1164, 1192, 1208

Offset: 1

Benoit Cloitre, Apr 30 2002

Georg Fischer, Table of n, a(n) for n = 1..2000 (first 1835 terms by Marius A. Burtea)

Cf. A000005, A000010, A009195.

[n: n in [1..2000] | GCD(n,EulerPhi(n)) eq NumberOfDivisors(n) ];// Marius A. Burtea, Dec 28 2018

Select[Range[1300], GCD[#, EulerPhi[#]] == DivisorSigma[0, #] &] (* Jayanta Basu, Mar 21 2013 *)

for(n=1,1592,if(gcd(n,eulerphi(n))==numdiv(n),print1(n,",")))

A074390 a(n) is the least number k that A074389(k) = n.

Original entry on oeis.org

1, 6, 18, 12, 200, 42, 196, 56, 459, 950, 5203, 396, 9243, 980, 1800, 336, 19363, 270, 13357, 600, 1764, 10406, 72473, 168, 18625, 34814, 4293, 812, 145493, 1350, 15376, 992, 19602, 38726, 41615, 1836, 99937, 26714, 1521, 440, 274003, 3822, 475193

Offset: 1

Labos Elemer, Aug 23 2002

nonn

			For n = 79: a(79) = 979837 because GCD(979837,998718,961272) = 79 and 979837 is the smallest.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000005, A000010, A073815, A050399, A009195, A009194.

f[x_] := GCD[DivisorSigma[1, x], EulerPhi[x], x]; t=Table[0, {100}]; Do[s=f[n]; If[s<101&&t[[s]]==0, t[[s]]=n], {n, 1, 1000000}]; t

lista(len) = {my(v = vector(len), c = 0, k = 1, f, i); while(c < len, f = factor(k); i = gcd([k, sigma(k), eulerphi(k)]); if(i <= len && v[i] == 0, c++; v[i] = k); k++); v;} \\ Amiram Eldar, Nov 14 2024

a(n) = Min{x; GCD(x, sigma(x), phi(x)) = n} = Min{x; GCD(x, A000005(x), A000010(x)) = n}.

A075857 Least common multiple of totient and cototient of n.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 6, 4, 6, 12, 10, 8, 12, 24, 56, 8, 16, 12, 18, 24, 36, 60, 22, 16, 20, 84, 18, 48, 28, 88, 30, 16, 260, 144, 264, 24, 36, 180, 120, 48, 40, 60, 42, 120, 168, 264, 46, 32, 42, 60, 608, 168, 52, 36, 120, 96, 252, 420, 58, 176, 60, 480, 108, 32, 816, 460, 66

Offset: 1

Reinhard Zumkeller, Oct 15 2002

nonn

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

Cf. A000010, A009195, A051953.

Table[LCM[EulerPhi[n],(n-EulerPhi[n])],{n,70}] (* Harvey P. Dale, Jun 10 2019 *)

a(n) = {my(p = eulerphi(n)); lcm(p, n-p);} \\ Amiram Eldar, Nov 30 2024

a(n) = A000010(n)*A051953(n)/A009195(n).

A082506 a(n) = gcd(2^n, n - phi(n)); largest power of 2 dividing cototient(n) = A051953(n).

Original entry on oeis.org

2, 1, 1, 2, 1, 4, 1, 4, 1, 2, 1, 8, 1, 8, 1, 8, 1, 4, 1, 4, 1, 4, 1, 16, 1, 2, 1, 16, 1, 2, 1, 16, 1, 2, 1, 8, 1, 4, 1, 8, 1, 2, 1, 8, 1, 8, 1, 32, 1, 2, 1, 4, 1, 4, 1, 32, 1, 2, 1, 4, 1, 32, 1, 32, 1, 2, 1, 4, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 4, 1, 4, 1, 16, 1, 2, 1, 16, 1, 16, 1, 64, 1, 8, 1

Offset: 1

Labos Elemer, Apr 28 2003

nonn

a(n)=1 if and only if n is odd or n = 2. - Robert Israel, May 31 2018

			Different from A069177, analogous sequence with totient, instead of cototient.

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

Cf. A000010, A051953, A009195, A083250, A007283, A050339, A053576, A069177.

f:= n -> padic:-ordp(n - numtheory:-phi(n), 2):
map(f, [$1..100]); # Robert Israel, May 31 2018

A306528 Numbers k such that gcd(k, phi(k)) = gcd(k, psi(k)).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 42, 43, 44, 46, 47, 49, 50, 52, 53, 56, 58, 59, 61, 62, 64, 65, 67, 68, 70, 71, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88, 89, 92, 94, 97, 98, 100

Offset: 1

Torlach Rush, Feb 21 2019

nonn

Here phi(n) is Euler's totient function A000010 and psi(n) is Dedekind's psi function A001615.
This sequence contains all prime powers p^k where phi(p^k) and psi(p^k) are equidistant from p^k, and gcd(p^k, phi(p^k)) = gcd(p^k, psi(p^k)) = p^(k - 1). For the prime numbers themselves this is trivial since phi(p) and psi(p) differ from p by 1 and 1^0 = 1.
If prime p|k, then p*k is in the sequence if and only if k is in the sequence. - Robert Israel, Mar 05 2019

			1 is a term because gcd(1, 1) = gcd(1, 1) = 1.
2 is a term because gcd(2, 1) = gcd(2, 3) = 1.
3 is a term because gcd(3, 2) = gcd(3, 4) = 1.
4 is a term because gcd(4, 2) = gcd(4, 6) = 2.
5 is a term because gcd(5, 4) = gcd(5, 6) = 1.
6 is not a term because gcd(6, 2) <> gcd(6, 12).
7 is a term because gcd(7, 6) = gcd(7, 8) = 1.

Cf. A000010, A001615, A009195, A306695.

filter:= proc(n) local p,F;
  F:= numtheory:-factorset(n);
  igcd(n, n*mul(1-1/p, p=F)) = igcd(n, n*mul(1+1/p,p=F))
end proc:
select(filter, [$1..200]); # Robert Israel, Mar 05 2019

dpsi(n) = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615
isok(k) = gcd(k, eulerphi(k)) == gcd(k, dpsi(k)); \\ Michel Marcus, Feb 27 2019

A326416 The numbers k for which gcd(k, phi(k)) + gcd(k, tau(k)) = gcd(k, sigma(k)).

Original entry on oeis.org

3040, 9440, 22240, 27360, 28640, 30080, 50560, 54288, 60640, 67040, 76752, 79840, 99040, 105440, 109888, 118240, 137440, 152960, 163040, 189072, 200160, 201440, 211536, 233440, 234880, 239840, 249216, 252640, 256128, 256464, 259040, 271840, 278928, 296320

Offset: 1

Marius A. Burtea, Oct 18 2019

nonn

The terms of the sequence are solutions of the equation A009191(k) + A009195(k) = A009194(k). All terms are composite numbers.

It seems that tau(a(n)) >= 24.

			For k = 3040 = 2^5 * 5 * 19, phi(k) = 2^4 * 4 * 18 = 2^7 * 3^2, tau(k) = 6* 2* 2 = 2^3 * 3, sigma(k) = 2^3 * 3^3 * 5 *7, gcd(k,phi(k)) + gcd(k tau(k)) = 2^5 + 2^3 = 40 and gcd(k,sigma(k)) = 2^3 * 5 = 40.

Cf. A000005, A000010, A000203, A009191, A009194, A009195, A328651.

[k: k in [1..300000]| Gcd(k,NumberOfDivisors(k))+Gcd(k,EulerPhi(k)) eq Gcd(k,SumOfDivisors(k))];

aQ[n_] := GCD[n, EulerPhi[n]] + GCD[n, DivisorSigma[0, n]] ==  GCD[n, DivisorSigma[1, n]]; Select[Range[300000], aQ] (* Amiram Eldar, Oct 23 2019 *)

isok(k) = gcd(k, numdiv(k)) + gcd(k, eulerphi(k)) == gcd(k, sigma(k)); \\ Michel Marcus, Oct 24 2019

A331176 a(n) = n - n/gcd(n, phi(n)), where phi is Euler totient function.

Original entry on oeis.org

0, 0, 0, 2, 0, 3, 0, 6, 6, 5, 0, 9, 0, 7, 0, 14, 0, 15, 0, 15, 14, 11, 0, 21, 20, 13, 24, 21, 0, 15, 0, 30, 0, 17, 0, 33, 0, 19, 26, 35, 0, 35, 0, 33, 30, 23, 0, 45, 42, 45, 0, 39, 0, 51, 44, 49, 38, 29, 0, 45, 0, 31, 56, 62, 0, 33, 0, 51, 0, 35, 0, 69, 0, 37, 60, 57, 0, 65, 0, 75, 78, 41, 0, 77, 0, 43, 0, 77, 0, 75

Offset: 1

Antti Karttunen, Jan 11 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000010, A009195, A109395.

Cf. A003277 (indices of zeros).

Table[n-n/GCD[n,EulerPhi[n]],{n,90}] (* Harvey P. Dale, Aug 20 2020 *)

A331176(n) = (n - (n/gcd(n, eulerphi(n))));

a(n) = n - A109395(n).

A362414 a(n) = gcd(n, phi(n)^2) / gcd(n, phi(n)).

Original entry on oeis.org

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

Offset: 1

Juri-Stepan Gerasimov, Apr 19 2023

a(n) = 1 if n is squarefree.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A000010, A000188, A009195.

[Gcd(n, EulerPhi(n)^2) / Gcd(n, EulerPhi(n)): n in [1..100]];

A362414[n_]:=With[{p=EulerPhi[n]},GCD[n,p^2]/GCD[n,p]];
Array[A362414,100] (* Paolo Xausa, Oct 22 2023 *)

a(n)=my(f=eulerphi(n)); gcd(n,f^2)/gcd(n,f) \\ Charles R Greathouse IV, May 03 2023

a(n) = gcd(n,A127473(n)) / A009195(n).

1 <= a(n) <= sqrt(n). The lower bound is sharp (squarefree numbers), as is the upper bound (squares of primes). - Charles R Greathouse IV, May 03 2023

cp's OEIS Frontend

A064448 a(n) = gcd(n^n, phi(n^n)).

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A068502 Composite numbers k such that gcd(sigma(k), k) = gcd(k, phi(k)).

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A069809 Numbers k such that gcd(k, phi(k)) = tau(k).

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A074390 a(n) is the least number k that A074389(k) = n.

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A075857 Least common multiple of totient and cototient of n.

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A082506 a(n) = gcd(2^n, n - phi(n)); largest power of 2 dividing cototient(n) = A051953(n).

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A306528 Numbers k such that gcd(k, phi(k)) = gcd(k, psi(k)).

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A326416 The numbers k for which gcd(k, phi(k)) + gcd(k, tau(k)) = gcd(k, sigma(k)).

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