A000010 -id:A000010 - cp's OEIS frontend

A239443 a(n) = phi(n^9), where phi = A000010.

1, 256, 13122, 131072, 1562500, 3359232, 34588806, 67108864, 258280326, 400000000, 2143588810, 1719926784, 9788768652, 8854734336, 20503125000, 34359738368, 111612119056, 66119763456, 305704134738, 204800000000, 453874312332, 548758735360, 1722841676182, 880602513408, 3051757812500

Offset: 1

José María Grau Ribas, Mar 22 2014

Number of solutions of the equation GCD(x_1^2 + ... + x_9^2,n)=1 with 0 < x_i <= n.

In general, for m>0, Sum_{k=1..n} phi(k^m) ~ 6 * n^(m+1) / ((m+1)*Pi^2). - Vaclav Kotesovec, Feb 02 2019

Seiichi Manyama, Table of n, a(n) for n = 1..10000
C. Calderón, J. M. Grau, A. Oller-Marcén, and László Tóth, Counting invertible sums of squares modulo n and a new generalization of Euler totient function, arXiv:1403.7878 [math.NT], 2014.

Defining Phi_k(n):= number of solutions of the equation GCD(x_1^2 + ... + x_k^2,n)=1 with 0 < x_i <= n.
Phi_1(n) = phi(n) = A000010(n).
Phi_2(n) = A079458(n).
Phi_3(n) = phi(n^3) = n^2*phi(n)= A053191(n).
Phi_4(n) = A227499(n).
Phi_5(n) = phi(n^5) = n^4*phi(n)= A238533(n).
Phi_6(n) = A238534(n).
Phi_7(n) = phi(n^7) = n^6*phi(n)= A239442(n).
Phi_8(n) = A239441(n).
Phi_9(n) = phi(n^9) = n^8*phi(n)= A239443(n).

with(numtheory); A239443:=n->phi(n^9); seq(A239443(n), n=1..100); # Wesley Ivan Hurt, Apr 01 2014

Mathematica
```
Table[EulerPhi[n^9], {n, 100}]
```

a(n) = n^8*eulerphi(n); \\ Michel Marcus, Mar 10 2018

Dirichlet g.f.: zeta(s - 9) / zeta(s - 8). The n-th term of the Dirichlet inverse is n^8 * A023900(n) = (-1)^omega(n) * a(n) / A003557(n), where omega = A001221. - Álvar Ibeas, Nov 24 2017
a(n) = n^8 * phi(n). - Altug Alkan, Mar 10 2018
Sum_{k=1..n} a(k) ~ 3*n^10 / (5*Pi^2). - Vaclav Kotesovec, Feb 02 2019
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p/(p^10 - p^9 - p + 1)) = 1.00399107654133714629... - Amiram Eldar, Dec 06 2020

A285702 a(n) = A000010(A064216(n)).

Original entry on oeis.org

1, 1, 2, 4, 2, 6, 10, 2, 12, 16, 4, 18, 6, 4, 22, 28, 6, 8, 30, 10, 36, 40, 4, 42, 20, 12, 46, 12, 16, 52, 58, 8, 20, 60, 18, 66, 70, 6, 24, 72, 8, 78, 24, 22, 82, 40, 28, 32, 88, 12, 96, 100, 8, 102, 106, 30, 108, 36, 20, 48, 42, 36, 18, 112, 40, 126, 64, 8, 130, 136, 42, 60, 44, 20, 138, 148, 24, 56, 150, 46, 72, 156, 12, 162, 110, 32, 166, 24, 52, 172, 178

Offset: 1

Antti Karttunen, Apr 26 2017

nonn

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

Cf. A000010, A064216, A064989, A099774, A151799, A285703.

Odd bisection of the following sequences: A347115, A348045, A349127, A349128.

Table[EulerPhi@ If[n == 1, 1, Apply[Times, FactorInteger[2 n - 1] /. {p_, e_} /; p > 2 :> NextPrime[p, -1]^e]], {n, 91}] (* Michael De Vlieger, Apr 26 2017 *)

(define (A285702 n) (A000010 (A064216 n)))

a(n) = A000010(A064216(n)).

Sum_{k=1..n} a(k) ~ c * n^2, where c = Product_{p prime} (p^3/((p+1)*(p^2-q(p)))) = 0.5366875995..., where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - Amiram Eldar, Dec 21 2023

A295301 a(n) = n - phi(sigma(n)), where phi = A000010 and sigma = A000203.

Original entry on oeis.org

0, 0, 1, -2, 3, 2, 3, 0, -3, 4, 7, 0, 7, 6, 7, -14, 11, -6, 11, 8, 5, 10, 15, 8, -5, 14, 11, 4, 21, 6, 15, -4, 17, 16, 19, -36, 19, 22, 15, 16, 29, 10, 23, 20, 21, 22, 31, -12, 13, -10, 27, 10, 35, 22, 31, 24, 25, 34, 43, 12, 31, 30, 15, -62, 41, 18, 35, 32, 37, 22, 47, -24, 37, 38, 15, 28, 45, 30, 47, 20

Offset: 1

Antti Karttunen, Nov 21 2017

sign

Antti Karttunen, Table of n, a(n) for n = 1..32768

Cf. A000010, A000203, A062401, A295303.
Cf. A001229 (positions of zeros), A066694 (of negative terms).
Cf. also A295302, A295305.

a:=List([1..80],n->n-Phi(Sigma(n)));; Print(a); # Muniru A Asiru, Jan 02 2019

[n-EulerPhi(SumOfDivisors(n)):n in [1..100]]; // Marius A. Burtea, Jan 01 2019

with(numtheory): seq(n-phi(sigma(n)),n=1..80); # Muniru A Asiru, Jan 02 2019

Array[# - EulerPhi@ DivisorSigma[1, #] &, 80] (* Michael De Vlieger, Jan 01 2019 *)

PARI
```
A295301(n) = (n - eulerphi(sigma(n)));
```

a(n) = n - A062401(n).

A324104 a(1) = 0; for n > 1, a(n) = A000010(A156552(n)).

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 4, 6, 2, 6, 8, 10, 16, 16, 4, 8, 32, 12, 64, 18, 6, 20, 128, 22, 4, 48, 6, 24, 256, 12, 512, 30, 16, 84, 8, 18, 1024, 256, 20, 24, 2048, 36, 4096, 66, 10, 324, 8192, 46, 8, 20, 48, 130, 16384, 28, 12, 70, 84, 800, 32768, 42, 65536, 1364, 18, 36, 32, 44, 131072, 216, 256, 40, 262144, 40, 524288, 3840, 12, 408, 16, 108

Offset: 1

Antti Karttunen, Feb 18 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2997

Cf. A000010, A156552, A324103.

Cf. also A323243, A324105 (sigma and tau similarly permuted).

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
A324104(n) = if(1==n,0,eulerphi(A156552(n)));

A330068 Numbers k such that Sum_{i=1..k} i^A000010(k) == 2 (mod k).

Original entry on oeis.org

1, 3, 4, 12, 84, 3612, 94116, 4429004844, 104990793204

Offset: 1

José María Grau Ribas, Nov 30 2019

Apparently includes the sequence 2*A054377.
Additional terms include 4429004844, 104990793204, and 16980843167119376821413542522172.
a(10) > 10^13. - Giovanni Resta, Feb 27 2020

Cf. A000010, A054377, A007850, A330069.

G[n_, k_] := G[n, k] = Mod[Sum[PowerMod[i, k, n], {i, 1, n}], n];
Select[Range[2000], G[#, EulerPhi[#]] == 2 &]
fa=FactorInteger;
se[n_, k_] := Select[Transpose[fa[n]][[1]], IntegerQ[k/(# - 1)] &];
sumlis[li_] := Sum[li[[i]], {i, 1, Length[li]}]
Table[If[Mod[-n/se[n, EulerPhi[n]] // sumlis, n] == 2, n], {n, 1, 1000000}] // Union

isok(n) = sumdiv(n, d, eulerphi(n/d) * Mod(d, n)^eulerphi(n)) == 2; \\ Daniel Suteu, Jan 13 2020

a(8)-a(9) from Giovanni Resta, Feb 27 2020

A330069 Numbers k such that Sum_{i=1..k} i^A000010(k) == -2 (mod k).

Original entry on oeis.org

1, 4, 60, 1716, 3444, 132396, 4428816612, 48846257124

Offset: 1

José María Grau Ribas, Nov 30 2019

Apparently includes the sequence 2*A007850.
Additional terms include 4428816612, 48846257124, 865498410347676, 29474266940021148, 1101686782618260636, 488394001964999430175732692, 1108159829234141602577157118356, 3821334362841015969111519832677012.
a(9) > 10^13. - Giovanni Resta, Feb 27 2020

Cf. A000010, A054377, A007850, A330068.

G[n_, k_] := G[n, k] = Mod[Sum[PowerMod[i, k, n], {i, 1, n}], n];
Select[Range[2000], G[#, EulerPhi[#]] == n-2 &]
fa=FactorInteger;
se[n_, k_] := Select[Transpose[fa[n]][[1]], IntegerQ[k/(# - 1)] &];
sumlis[li_] := Sum[li[[i]], {i, 1, Length[li]}]
Table[If[Mod[-n/se[n, EulerPhi[n]] // sumlis, n] == n-2, n], {n, 1,
   1000000}] // Union

isok(n) = sumdiv(n, d, eulerphi(n/d) * Mod(d, n)^eulerphi(n)) == -2; \\ Daniel Suteu, Jan 13 2020

a(7)-a(8) from Giovanni Resta, Feb 27 2020

A053478 Sum of iterates when phi, A000010, is iterated until fixed point 1.

Original entry on oeis.org

1, 3, 6, 7, 12, 9, 16, 15, 18, 17, 28, 19, 32, 23, 30, 31, 48, 27, 46, 35, 40, 39, 62, 39, 60, 45, 54, 47, 76, 45, 76, 63, 68, 65, 74, 55, 92, 65, 78, 71, 112, 61, 104, 79, 84, 85, 132, 79, 110, 85, 114, 91, 144, 81, 126, 95, 112, 105, 164, 91, 152, 107, 118, 127, 144, 101

Offset: 1

Labos Elemer, Jan 14 2000

nonn

For n = 2^w, the sum is 2^(w+1) - 1.

			If phi is applied repeatedly to n = 91, the iterates {91, 72, 24, 8, 4, 2, 1} are obtained. Their sum is a(91) = 91 + 72 + 24 + 8 + 4 + 2 + 1 = 202.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000010, A010554, A049099, A049100, A049107, A049108, A032358.

Row sums of A375478.

a053478 = (+ 1) . sum . takeWhile (/= 1) . iterate a000010
-- Reinhard Zumkeller, Oct 27 2011

f[n_] := Plus @@ Drop[ FixedPointList[ EulerPhi, n], -1]; Table[ f[n], {n, 66}] (* Robert G. Wilson v, Dec 16 2004 *)
f[1] := 1; f[n_] := n + f[EulerPhi[n]]; Table[f[n], {n, 66}] (* Carlos Eduardo Olivieri, May 26 2015 *)

a(n)=my(s=n);while(n>1,s+=n=eulerphi(n)); s \\ Charles R Greathouse IV, Feb 21 2013

a(n) = n + a(phi(n)).

a(n) = A092693(n) + n. - Vladeta Jovovic, Jul 02 2004

A078892 Numbers n such that phi(n) - 1 is prime, where phi is Euler's totient function (A000010).

Original entry on oeis.org

5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 24, 25, 26, 27, 28, 30, 31, 33, 35, 36, 38, 39, 42, 43, 44, 45, 49, 50, 51, 52, 54, 56, 61, 62, 64, 65, 66, 68, 69, 70, 72, 73, 77, 78, 80, 81, 84, 86, 90, 91, 92, 93, 95, 96, 98, 99, 102, 103, 104, 105, 109, 111, 112, 117

Offset: 1

Reinhard Zumkeller, Dec 12 2002

For all primes p: p is in the sequence iff p is the greater member of a twin prime pair (A006512), see A078893.

Union of A006512 and A078893. - Ray Chandler, May 26 2008

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000010, A000040, A008864, A109606, A006512, A039698, A072281, A078893.

[n: n in [1..200] | IsPrime(EulerPhi(n)-1)]; // Vincenzo Librandi, Aug 13 2013

Select[Range[200], PrimeQ[EulerPhi[#] - 1]&] (* Vincenzo Librandi, Aug 13 2013 *)

is(n)=isprime(eulerphi(n)-1) \\ Charles R Greathouse IV, Feb 21 2013

A112955 Greatest number m such that phi(m) = n*tau(m), with phi=A000010 and tau=A000005; a(n)=0 if no such m exists.

Original entry on oeis.org

30, 120, 210, 420, 330, 840, 294, 1260, 1080, 1320, 690, 2520, 318, 1470, 2310, 3360, 0, 3780, 0, 4620, 1290, 2760, 1410, 5460, 3000, 1590, 7560, 5880, 1770, 9240, 0, 10080, 4830, 1236, 3234, 10920, 894, 0, 2370, 13860, 2490, 6090, 1038, 9660, 11880

Offset: 1

Reinhard Zumkeller, Oct 07 2005

nonn

Each term is a multiple of 6. [Max Alekseyev, Mar 01 2010]

			a(1) = A020488(A112954(1)) = A020488(7) = 30;
a(2) = A062516(A112954(2)) = A062516(9) = 120;
a(3) = A063469(A112954(3)) = A063469(10) = 210;
a(4) = A063470(A112954(4)) = A063470(9) = 420.

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A112954.

Cf. A175667. [Enrique Pérez Herrero, Oct 22 2010]

More terms from Max Alekseyev, Mar 01 2010

A130207 Diagonalized matrix of A000010, Euler totient function phi.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6

Offset: 1

Gary W. Adamson, May 16 2007

			First few rows of the triangle are:
1;
0, 1;
0, 0, 2;
0, 0, 0, 2;
0, 0, 0, 0, 4;
...

Antti Karttunen, Table of n, a(n) for n = 1..22155; the first 210 rows of the triangle

Cf. A000010, A130208, A130209.

A130207 := proc(n,k)
    if k = n then
        numtheory[phi](n);
    else
        0;
    end if;
end proc:
seq(seq(A130207(n,k),k=1..n),n=1..15) ;

for(n=1,9,for(k=2,n,print1("0, "));print1(eulerphi(n)", ")) \\ Charles R Greathouse IV, Feb 19 2013

A130207(n) = if(ispolygonal(n,3), eulerphi((sqrtint(1+(n*8))-1)/2), 0); \\ Antti Karttunen, Jan 17 2025

T(n,n) = A000010(n).

T(n,k) = 0, if k <> n.

Data section extended up to a(105) by Antti Karttunen, Jan 17 2025

cp's OEIS Frontend

A239443 a(n) = phi(n^9), where phi = A000010.

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A285702 a(n) = A000010(A064216(n)).

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A295301 a(n) = n - phi(sigma(n)), where phi = A000010 and sigma = A000203.

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A324104 a(1) = 0; for n > 1, a(n) = A000010(A156552(n)).

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A330068 Numbers k such that Sum_{i=1..k} i^A000010(k) == 2 (mod k).

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A330069 Numbers k such that Sum_{i=1..k} i^A000010(k) == -2 (mod k).

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A053478 Sum of iterates when phi, A000010, is iterated until fixed point 1.

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A078892 Numbers n such that phi(n) - 1 is prime, where phi is Euler's totient function (A000010).

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