A014573 -id:A014573 - cp's OEIS frontend

A090555 Erroneous version of A014573.

10, 1, 2, 4, 8, 16

Offset: 1

dead

A007374 Smallest k such that phi(x) = k has exactly n solutions, n>=2.

1, 2, 4, 8, 12, 32, 36, 40, 24, 48, 160, 396, 2268, 704, 312, 72, 336, 216, 936, 144, 624, 1056, 1760, 360, 2560, 384, 288, 1320, 3696, 240, 768, 9000, 432, 7128, 4200, 480, 576, 1296, 1200, 15936, 3312, 3072, 3240, 864, 3120, 7344, 3888, 720, 1680, 4992

Offset: 2

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

The Carmichael Totient Conjecture is that there is no k such that phi(x) = k has a unique solution x. So a(1) does not exist.

Ford proved that a(n) exists for all n > 1. - Charles R Greathouse IV, Oct 13 2014

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
Wacław Sierpiński, Elementary Theory of Numbers, p. 234, Warsaw, 1964.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jud McCranie, Table of n, a(n) for n = 2..10000 (terms up to 1..1023 from T. D. Noe, terms 1024..7448 from Donovan Johnson)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Kevin Ford, The distribution of totients, Ramanujan J., (2) No. 1-2 (1998); New version of the 1998 article, arXiv:1104.3264 [math.NT], 2011-2013.
Kevin Ford, The number of solutions of phi(x) = m, Annals of Mathematics 150:1 (1999), pp. 283-311.
S. D. Merow, Has Carmichael's Totient Conjecture Been Proven? No, No, It Has Not, Notices Amer. Math. Soc., 66 (No. 5, 2019), 759-761.
A. Schlafly and S. Wagon, Carmichael's conjecture on the Euler function is valid below 10^{10,000,000}, Mathematics of Computation, 63 No. 207 (1994), 415-419. See Table 2.
Eric Weisstein's World of Mathematics, Phi function.
Eric Weisstein's World of Mathematics, Carmichael's Totient Function Conjecture.

Essentially same as A014573. Records in A105207, A105208.

Cf. A000010, A097942, A105207, A105208.

a = Table[ 0, {10^5} ]; Do[ s = EulerPhi[ n ]; If[ s < 100001, a[ [ s ] ]++ ], {n, 1, 10^6} ]; Do[ k = 1; While[ a[ [ k ] ] != n, k++ ]; Print[ k ], {n, 2, 75} ]

v=vectorsmall(10^6);for(n=1,1e7,t=eulerphi(n);if(t<=#v,v[t]++))
u=vector(100);for(i=1,#v,t=v[i];if(t&&t<=#u&&u[t]==0,u[t]=i)); u[2..#u]
\\ Charles R Greathouse IV, Oct 13 2014

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

A072075 Smallest solution to phi(x) = 10^n where phi(x) = A000010(x).

Original entry on oeis.org

1, 11, 101, 1111, 10291, 100651, 1004251, 10165751, 100064101, 1000078501, 10000222501, 100062501601, 1000062516001, 10000062660001, 100002441447211, 1003922328562757, 10000390625025601, 100000002482366251, 1000000002851006251, 10000062500000160001

Offset: 0

Labos Elemer, Jun 13 2002

nonn

			n=3: a(3)=1111 because InvPhi[1000]= {1111,1255,1375,1875,2008,2222,2500,2510,2750,3012,3750}.

Ray Chandler, Table of n, a(n) for n = 0..1000
Max A. Alekseyev, Computing the Inverses, their Power Sums, and Extrema for Euler's Totient and Other Multiplicative Functions. Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.2

Cf. A000010, A014573, A014197, A110077, A072074, A072076.

More terms from Max Alekseyev, Apr 26 2010

a(18)-a(19) from Donovan Johnson, Feb 02 2012

A072074 Number of integers k such that phi(k) = 10^n.

Original entry on oeis.org

2, 2, 4, 11, 16, 24, 43, 63, 94, 152, 224, 324, 464, 644, 897, 1271, 1790, 2521, 3501, 4814, 6535, 8779, 11739, 15585, 20625, 27166, 35588, 46363, 60065, 77424, 99337, 127020, 161930, 205847, 260929, 329782, 415533, 522173, 654548, 818278, 1020391

Offset: 0

Labos Elemer, Jun 13 2002

nonn

a(n) is the coefficient of x^n*y^n in Product_p Sum_{u, v} x^u*y^v, where the product is taken over all primes p and the sum is taken over such u, v that 2^u*5^v = phi(p^k) for some nonnegative integer k. - Max Alekseyev, Apr 26 2010

Elaborating on above comment, primes p must be in A077497 and k must be 1 for primes other than 2 and 5. - Ray Chandler, Feb 12 2012

			n=3: a(3)=11 because InvPhi(1000) = {1111, 1255, 1375, 1875, 2008, 2222, 2500, 2510, 2750, 3012, 3750}.

Ray Chandler, Table of n, a(n) for n = 0..1000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Max A. Alekseyev, Computing the Inverses, their Power Sums, and Extrema for Euler's Totient and Other Multiplicative Functions. Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.2.

Cf. A000010, A014197, A014573, A110078, A072075, A072076.

Maple
```
[seq(nops(invphi(10^i)),i=1..8)];
```

a(n) = #invphi(10^n); \\ for invphi see Alekseyev link \\ Michel Marcus, May 14 2020

a(n) = Card{x : A000010(x)=10^n}.

More terms from Max Alekseyev, Apr 26 2010

A072076 Largest k such that EulerPhi(k) = 10^n.

Original entry on oeis.org

2, 22, 250, 3750, 41250, 414150, 4166250, 42281250, 438281250, 4400343750, 44266406250, 449238281250, 4510352343750, 45373066406250, 455545586718750, 4555455867187500, 45555287544813750, 455552875448137500, 4566844506855468750, 45668445068554687500

Offset: 0

Labos Elemer, Jun 13 2002

nonn

			n=3: a(3)=3750 because InvPhi(1000) = {1111, 1255, 1375, 1875, 2008, 2222, 2500, 2510, 2750, 3012, 3750}.

Ray Chandler, Table of n, a(n) for n = 0..1000
Max A. Alekseyev, Computing the Inverses, their Power Sums, and Extrema for Euler's Totient and Other Multiplicative Functions. Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.2

Cf. A000010, A014197, A014573, A072074, A072075, A110076.

a(n) = Max{k; A000010(k) = 10^n}.

More terms from Max Alekseyev, Apr 26 2010

A090556 Beginning with 1, a(n) = least number m > a(n-1) such that phi(a(n-1)) divides phi(m).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 10, 12, 13, 21, 26, 28, 35, 39, 45, 52, 56, 65, 97, 119, 153, 193, 221, 291, 357, 386, 388, 416, 442, 448, 476, 485, 579, 595, 663, 765, 769, 965, 1105, 1455, 1538, 1544, 1552, 1649, 1799, 2307, 2313, 2895, 3076, 3088, 3104, 3281, 3341, 3845, 4947

Offset: 1

Amarnath Murthy, Dec 10 2003

nonn

Ivan Neretin, Table of n, a(n) for n = 1..308

Cf. A014573, A090557.

NestList[Function[n, SelectFirst[Range[n + 1, 10^4], Divisible[EulerPhi@ #, EulerPhi@ n] &]], 1, 54] (* Michael De Vlieger, May 01 2016, Version 10 *)

a(1)=1, then a(n+1) = A069797(a(n)). - Ivan Neretin, May 01 2016

Corrected and extended by Sam Handler (sam_5_5_5_0(AT)yahoo.com), Nov 03 2004

A090557 a(n) = phi(A090556(n)).

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 4, 4, 12, 12, 12, 12, 24, 24, 24, 24, 24, 48, 96, 96, 96, 192, 192, 192, 192, 192, 192, 192, 192, 192, 192, 384, 384, 384, 384, 384, 768, 768, 768, 768, 768, 768, 768, 1536, 1536, 1536, 1536, 1536, 1536, 1536, 1536, 3072, 3072, 3072, 3072, 3072

Offset: 1

Amarnath Murthy, Dec 10 2003

nonn

Starting at n=9, all known terms are of the form 3*2^k. - Ivan Neretin, May 01 2016

Ivan Neretin, Table of n, a(n) for n = 1..308

Cf. A014573, A090556.

EulerPhi@ NestList[Function[n, SelectFirst[Range[n + 1, 10^4], Divisible[EulerPhi@ #, EulerPhi@ n] &]], 1, 55] (* Michael De Vlieger, May 01 2016, Version 10 *)

More terms from David Wasserman, Jan 04 2006

Missing term a(1) prepended by Ivan Neretin, May 01 2016

cp's OEIS Frontend

A090555 Erroneous version of A014573.

Views

Author

Keywords

A007374 Smallest k such that phi(x) = k has exactly n solutions, n>=2.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

Extensions

A072075 Smallest solution to phi(x) = 10^n where phi(x) = A000010(x).

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A072074 Number of integers k such that phi(k) = 10^n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula

Extensions

A072076 Largest k such that EulerPhi(k) = 10^n.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

A090556 Beginning with 1, a(n) = least number m > a(n-1) such that phi(a(n-1)) divides phi(m).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A090557 a(n) = phi(A090556(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions