A000010 -id:A000010 - cp's OEIS frontend

A070812 a(n) = phi(gpf(n)) - gpf(phi(n)) = A000010(A006530(n)) - A006530(A000010(n)).

0, -1, 2, 0, 3, -1, -1, 2, 5, 0, 9, 3, 2, -1, 14, -1, 15, 2, 3, 5, 11, 0, -1, 9, -1, 3, 21, 2, 25, -1, 5, 14, 3, -1, 33, 15, 9, 2, 35, 3, 35, 5, 1, 11, 23, 0, -1, -1, 14, 9, 39, -1, 5, 3, 15, 21, 29, 2, 55, 25, 3, -1, 9, 5, 55, 14, 11, 3, 63, -1, 69, 33, -1, 15, 5, 9, 65, 2, -1, 35, 41, 3, 14, 35, 21, 5, 77, 1, 9, 11, 25, 23, 15, 0, 93, -1, 5

Offset: 3

Labos Elemer, May 09 2002

sign

Value of commutator[A000010, A006530] at n.

			Cases of n when a(n) = 1, -1, 2 or 0 are listed in A070002, A070003, A070004, A007283 respectively. Further regular solutions: if a(n)=3, then n=7k, where k has prime divisors < 7; if a(n)=5, then n=11k, where k has no prime divisors >=11; if a(n)=25, then mostly (not always!) n=31k ...

Michael De Vlieger, Table of n, a(n) for n = 3..10000
Michael De Vlieger, Log log scatterplot of a(n)+2, n = 3..10^5.

Cf. A000010, A006530, A070777, A068211, A070002, A070003, A070004, A007283.

pf[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] Table[EulerPhi[pf[u]]-pf[EulerPhi[u]], {u, 3, 128}]

gpf(n)=my(f=factor(n)[,1]);f[#f]
a(n)=gpf(n)-gpf(eulerphi(n))-1 \\ Charles R Greathouse IV, Feb 19 2013

a(n) = A070777(n) - A068211(n).

A083533 First difference sequence of A002202. Difference between consecutive possible values of phi(n), the Euler totient function A000010.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 2, 2, 4, 4, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 2, 4, 2, 6, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 4, 4, 6, 2, 2, 2, 4, 2, 2, 4, 4, 2, 6, 4, 2, 2, 2, 2, 4, 4, 2, 2, 4, 6, 2, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 2, 2, 4, 6, 2, 10, 2, 4, 4, 2, 2, 4, 2, 2, 4, 4, 2, 6, 4, 2, 2, 4, 6, 4, 2, 4

Offset: 1

Labos Elemer, May 20 2003

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000010, A002202, A005277, A083531, A083532, A083534, A083535, A083536.

a083533 n = a083533_list !! (n-1)
a083533_list = zipWith (-) (tail a002202_list) a002202_list
-- Reinhard Zumkeller, Nov 26 2015

t=Table[EulerPhi[w], {w, 1, 25000}]; u=Union[%]; Delete[u-RotateRight[u], 1]

lista(lim) = {my(k1 = 1, k2 = 1); while(k1 < lim, until(istotient(k2), k2++); print1(k2 - k1, ", "); k1 = k2);} \\ Amiram Eldar, Nov 16 2024

a(n) = A002202(n+1) - A002202(n).

A097942 Highly totient numbers: each number k on this list has more solutions to the equation phi(x) = k than any preceding k (where phi is Euler's totient function, A000010).

Original entry on oeis.org

1, 2, 4, 8, 12, 24, 48, 72, 144, 240, 432, 480, 576, 720, 1152, 1440, 2880, 4320, 5760, 8640, 11520, 17280, 25920, 30240, 34560, 40320, 51840, 60480, 69120, 80640, 103680, 120960, 161280, 181440, 207360, 241920, 362880, 483840, 725760, 967680

Offset: 1

Alonso del Arte, Sep 05 2004

nonn

If you inspect PhiAnsYldList after running the Mathematica program below, the zeros with even-numbered indices should correspond to the nontotients (A005277).
Where records occur in A014197. - T. D. Noe, Jun 13 2006
Cf. A131934.

			a(4) = 8 since phi(x) = 8 has the solutions {15, 16, 20, 24, 30}, one more solution than a(3) = 4 for which phi(x) = 4 has solutions {5, 8, 10, 12}.

Jud McCranie, Table of n, a(n) for n = 1..109 (terms 1..79 from T. D. Noe, terms 80..86 from Donovan Johnson)
Wikipedia, Highly totient number.

A subsequence of A007374.

Cf. A000010, A005277, A014573, A004653, A105207, A105208.

HighlyTotientNumbers := proc(n) # n > 1 is search maximum
local L, m, i, r; L := NULL; m := 0;
for i from 1 to n do
  r := nops(numtheory[invphi](i));
  if r > m then L := L,[i,r]; m := r fi
od; [L] end:
A097942_list := n -> seq(s[1], s = HighlyTotientNumbers(n));
A097942_list(500); # Peter Luschny, Sep 01 2012

searchMax = 2000; phiAnsYldList = Table[0, {searchMax}]; Do[phiAns = EulerPhi[m]; If[phiAns <= searchMax, phiAnsYldList[[phiAns]]++ ], {m, 1, searchMax^2}]; highlyTotientList = {1}; currHigh = 1; Do[If[phiAnsYldList[[n]] > phiAnsYldList[[currHigh]], highlyTotientList = {highlyTotientList, n}; currHigh = n], {n, 2, searchMax}]; Flatten[highlyTotientList]

{ A097942_list(n) = local(L, m, i, r);
  m = 0;
  for(i=1, n,
\\ from Max Alekseyev, http://home.gwu.edu/~maxal/gpscripts/
   r = numinvphi(i);
   if(r > m, print1(i,", "); m = r) );
} \\ Peter Luschny, Sep 01 2012

def HighlyTotientNumbers(n) : # n > 1 is search maximum.
    R = {}
    for i in (1..n^2) :
        r = euler_phi(i)
        if r <= n :
            R[r] = R[r] + 1 if r in R else 1
    # print R.keys()   # A002202
    # print R.values() # A058277
    P = []; m = 1
    for l in sorted(R.keys()) :
        if R[l] > m : m = R[l]; P.append((l,m))
    # print [l[0] for l in P] # A097942
    # print [l[1] for l in P] # A131934
    return P
A097942_list = lambda n: [s[0] for s in HighlyTotientNumbers(n)]
A097942_list(500) # Peter Luschny, Sep 01 2012

Edited and extended by Robert G. Wilson v, Sep 07 2004

A179188 Numbers n such that phi(n) = phi(n+6), with Euler's totient function phi=A000010.

Original entry on oeis.org

24, 34, 36, 39, 43, 44, 57, 72, 78, 82, 84, 93, 96, 108, 111, 146, 178, 201, 216, 222, 225, 226, 228, 306, 327, 364, 366, 381, 399, 417, 432, 438, 442, 466, 471, 482, 516, 527, 540, 543, 562, 576, 597, 610, 626, 633, 648, 706, 714, 732, 738, 802, 818, 866, 898, 912, 921, 924, 942, 948, 972, 1011

Offset: 1

M. F. Hasler, Jan 05 2011

nonn

There are 1385502728 terms under 10^12. - Jud McCranie, Feb 13 2012

Jud McCranie, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
F. Firoozbakht, Puzzle 466. phi(n-1)=phi(n)=phi(n+1), in C. Rivera's Primepuzzles.
Kevin Ford, Solutions of phi(n)=phi(n+k) and sigma(n)=sigma(n+k), arXiv:2002.12155 [math.NT], 2020.

Cf. A000010, A001274, A001494, A179186, A179187, A007015.

[n: n in [1..1000] | EulerPhi(n) eq EulerPhi(n+6)]; // Vincenzo Librandi, Sep 08 2016

Flatten[Position[Partition[EulerPhi[Range[1200]],7,1],?(#[[1]] == #[[7]]&),{1},Heads->False]] (* _Harvey P. Dale, Jan 30 2016 *)
Select[Range[1000], EulerPhi[#] == EulerPhi[# + 6] &] (* Vincenzo Librandi, Sep 08 2016 *)

{op=vector(N=6); for( n=1,1e4,if( op[n%N+1]+0==op[n%N+1]=eulerphi(n),print1(n-N,",")))}

A000010(a(n)) = A000010(a(n)+6).

A235137 a(n) = Sum_{k = 1..n} k^phi(n), where phi(n) = A000010(n).

Original entry on oeis.org

1, 3, 14, 30, 979, 91, 184820, 8772, 978405, 25333, 40851766526, 60710, 36720042483591, 19092295, 5666482312, 9961449608, 76762718946972480009, 105409929, 164309788542828686799730, 70540730666, 15909231318568907, 67403375450475, 1433191209985108404653810959324, 351625763020, 15975648280734359596251725645

Offset: 1

Jonathan Sondow and Emmanuel Tsukerman, Jan 03 2014

nonn

a(n) == -1 (mod n) if and only if n is prime or is a Giuga number A007850.

a(n) == 1 (mod n) if (and probably only if) n is a primary pseudoperfect number A054377.

			a(4) = 30 since 1^(phi(4)) + 2^(phi(4)) + 3^(phi(4)) + 4^(phi(4))= 1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = 30.
a(5) = 979, since phi(5) = 4 and 1^4 + 2^4 + 3^4 + 4^4 + 5^4 = 1 + 16 + 81 + 256 + 625 = 979.
a(6) = 91, since phi(6) = 2 and 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 1 + 4 + 9 + 16 + 25 + 36 = 91.

Seiichi Manyama, Table of n, a(n) for n = 1..388
J. Sondow and K. MacMillan, Reducing the Erdős-Moser equation 1^n + 2^n + ... + k^n = (k+1)^n modulo k and k^2, Integers 11 (2011), #A34.
J. Sondow and E. Tsukerman, The p-adic order of power sums, the Erdos-Moser equation, and Bernoulli numbers, arXiv:1401.0322 [math.NT], 2014; see section 4.
Wikipedia, Giuga number
Wikipedia, Primary pseudoperfect number

Cf. A000010, A007850, A054377, A235138.

a[n_] := Sum[PowerMod[i, EulerPhi@n, n], {i, n}]

a(n) = sum(k=1, n , k^eulerphi(n)); \\ Michel Marcus, Oct 21 2015

a(n) (mod n) = A235138(n).

A295502 a(n) = phi(5^n-1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

2, 8, 60, 192, 1400, 4320, 39060, 119808, 894240, 2912000, 24414060, 62208000, 610351560, 1959874560, 13154400000, 44043337728, 380537036928, 997843069440, 9485297382000, 25606963200000, 230106651919200, 748687423334400, 5959800062798400, 15138938880000000

Offset: 1

Seiichi Manyama, Nov 22 2017

nonn

Faye et al. prove that no term is of the form 5^k-1. - Michel Marcus, Jun 16 2024

Max Alekseyev, Table of n, a(n) for n = 1..502
Bernadette Faye, Florian Luca, and Amadou Tall, On the equation phi(5^m-1)=5^n-1, Bull. Korean Math. Soc. 2015; 52(2): 513-524.
Eric Weisstein's World of Mathematics, Totient Function
Wikipedia, Euler's totient function

Cf. A000010, A024049.

phi(k^n-1): A053287 (k=2), A295500 (k=3), A295501 (k=4), this sequence (k=5), A366623 (k=6), A366635 (k=7), A366654 (k=8), A366663 (k=9), A295503 (k=10), A366685 (k=11), A366711 (k=12).

EulerPhi[5^Range[25] - 1] (* Paolo Xausa, Jun 18 2024 *)

PARI
```
{a(n) = eulerphi(5^n-1)}
```

a(n) = n*A027741(n).

a(n) = A000010(A024049(n)). - Michel Marcus, Jun 16 2024

A319340 Sum of Euler totient function and its Dirichlet inverse: a(n) = A000010(n) + A023900(n).

Original entry on oeis.org

2, 0, 0, 1, 0, 4, 0, 3, 4, 8, 0, 6, 0, 12, 16, 7, 0, 8, 0, 12, 24, 20, 0, 10, 16, 24, 16, 18, 0, 0, 0, 15, 40, 32, 48, 14, 0, 36, 48, 20, 0, 0, 0, 30, 32, 44, 0, 18, 36, 24, 64, 36, 0, 20, 80, 30, 72, 56, 0, 8, 0, 60, 48, 31, 96, 0, 0, 48, 88, 0, 0, 26, 0, 72, 48, 54, 120, 0, 0, 36, 52, 80, 0, 12, 128, 84, 112, 50, 0, 16

Offset: 1

Antti Karttunen, Sep 17 2018

nonn

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

Cf. A000010, A023900, A051953, A318833, A319341.

A023900(n) = factorback(apply(p -> 1-p, factor(n)[, 1]));
A319340(n) = (eulerphi(n)+A023900(n));

a(n) = A000010(n) + A023900(n).

a(n) = A318833(n) - A051953(n).

A366635 a(n) = phi(7^n-1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

2, 16, 108, 640, 5600, 36288, 264992, 1536000, 12387168, 85120000, 658519752, 3135283200, 32296336800, 216063877120, 1450340640000, 8333819904000, 77537969371008, 488237947481088, 3790563394976072, 19162214400000000, 170264753751665664, 1245495178700551680

Offset: 1

Sean A. Irvine, Oct 15 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..388

phi(k^n-1): A053287 (k=2), A295500 (k=3), A295501 (k=4), A295502 (k=5), A366623 (k=6), this sequence (k=7), A366654 (k=8), A366663 (k=9), A295503 (k=10), A366685 (k=11), A366711 (k=12).

Cf. A000010, A024075, A366632, A366633, A366634.

EulerPhi[7^Range[30] - 1] (* Wesley Ivan Hurt, Oct 15 2023 *)

PARI
```
{a(n) = eulerphi(7^n-1)}
```

A366654 a(n) = phi(8^n-1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

6, 36, 432, 1728, 27000, 139968, 1778112, 6635520, 113467392, 534600000, 6963536448, 26121388032, 465193834560, 2427720325632, 28548223200000, 109586090557440, 1910296842179040, 9618417501143040, 123523151337020736, 406467072000000000, 7713001620195508224

Offset: 1

Sean A. Irvine, Oct 15 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..500

phi(k^n-1): A053287 (k=2), A295500 (k=3), A295501 (k=4), A295502 (k=5), A366623 (k=6), A366635 (k=7), this sequence (k=8), A366663 (k=9), A295503 (k=10), A366685 (k=11), A366711 (k=12).

Cf. A000010, A010705, A024088, A057953, A059890, A274908, A366651, A366652, A366653.

Mathematica
```
EulerPhi[8^Range[30] - 1]
```
PARI
```
{a(n) = eulerphi(8^n-1)}
```

from sympy import totient
def A366654(n): return totient((1<<3*n)-1) # Chai Wah Wu, Oct 15 2023

a(n) = A053287(3*n). - Max Alekseyev, Jan 09 2024

A366663 a(n) = phi(9^n-1), where phi is Euler's totient function (A000010).

Original entry on oeis.org

4, 32, 288, 2560, 26400, 165888, 2384928, 15728640, 141087744, 1246080000, 14758128000, 85996339200, 1270928131200, 8810420097024, 70207948800000, 677066362060800, 8218041445152000, 43129128265187328, 674757689572915200, 4238841176064000000

Offset: 1

Sean A. Irvine, Oct 15 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..690

phi(k^n-1): A053287 (k=2), A295500 (k=3), A295501 (k=4), A295502 (k=5), A366623 (k=6), A366635 (k=7), A366654 (k=8), this sequence (k=9), A295503 (k=10), A366685 (k=11), A366711 (k=12).

Cf. A000010, A024101, A057952, A366660, A366661, A366662, A366667.

Mathematica
```
EulerPhi[9^Range[30] - 1]
```
PARI
```
{a(n) = eulerphi(9^n-1)}
```

a(n) = A295500(2*n) = 2 * A295500(n) * A366579(n). - Max Alekseyev, Jan 07 2024

cp's OEIS Frontend

A070812 a(n) = phi(gpf(n)) - gpf(phi(n)) = A000010(A006530(n)) - A006530(A000010(n)).

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A083533 First difference sequence of A002202. Difference between consecutive possible values of phi(n), the Euler totient function A000010.

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A097942 Highly totient numbers: each number k on this list has more solutions to the equation phi(x) = k than any preceding k (where phi is Euler's totient function, A000010).

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A179188 Numbers n such that phi(n) = phi(n+6), with Euler's totient function phi=A000010.

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Magma

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A235137 a(n) = Sum_{k = 1..n} k^phi(n), where phi(n) = A000010(n).

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Mathematica

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A295502 a(n) = phi(5^n-1), where phi is Euler's totient function (A000010).

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Mathematica

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A319340 Sum of Euler totient function and its Dirichlet inverse: a(n) = A000010(n) + A023900(n).

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