A014197 -id:A014197 - cp's OEIS frontend

A048855 Number of integers up to n! relatively prime to n!.

1, 1, 1, 2, 8, 32, 192, 1152, 9216, 82944, 829440, 8294400, 99532800, 1194393600, 16721510400, 250822656000, 4013162496000, 64210599936000, 1155790798848000, 20804234379264000, 416084687585280000, 8737778439290880000, 192231125664399360000

Offset: 0

Paul Max Payton

Rephrasing the Quet formula: Begin with 1. Then, if n + 1 is prime subtract 1 and multiply. If n+1 is not prime, multiply. Continue writing each product. Thus the sequence would begin 1, 2, 8, . . . . The first product is 1*(2 - 1), second is 1*(3 - 1), and third is 2*4. - Enoch Haga, May 06 2009

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, A Foundation for Computer Science, Addison-Wesley Publ. Co., Reading, MA, 1989, page 134.

Charles R Greathouse IV, Table of n, a(n) for n = 0..450
Jean-Marie De Koninck and William Verreault, Arithmetic functions at factorial arguments, Publications de l'Institut Mathématique, Vol. 115, No. 129 (2024), pp. 45-76.

Cf. A000010, A000142, A014197.

Cf. A001620, A080130.

with(numtheory):a:=n->phi(n!): seq(a(n), n=0..20); # Zerinvary Lajos, Oct 07 2007

Table[ EulerPhi[ n! ], {n, 0, 21}] (* Robert G. Wilson v, Nov 21 2003 *)

a(n)=eulerphi(n!) \\ Charles R Greathouse IV, May 12 2011

from math import factorial, prod
from sympy import primerange
from fractions import Fraction
def A048855(n): return (factorial(n)*prod(Fraction(p-1,p) for p in primerange(n+1))).numerator # Chai Wah Wu, Jul 06 2022

[euler_phi(factorial(n)) for n in range(0,21)] # Zerinvary Lajos, Jun 06 2009

a(n) = phi(n!) = A000010(n!).
If n is composite, then a(n) = a(n-1)*n. If n is prime, then a(n) = a(n-1)*(n-1). - Leroy Quet, May 24 2007
Under the Riemann Hypothesis, a(n) = n! / (e^gamma * log n) * (1 + O(log n/sqrt(n))). - Charles R Greathouse IV, May 12 2011
Sum_{k=1..n} a(k) = exp(-gamma) * (n!/log(n)) * (1 + O(1/log(n)^3)), where gamma is Euler's constant (A001620) (De Koninck and Verreault, 2024, p. 56, eq. (4.12)). - Amiram Eldar, Dec 10 2024

Name changed by Daniel Forgues, Aug 01 2011

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

A120963 Number of monic polynomials with integer coefficients of degree n with all roots on the unit circle; number of products of cyclotomic polynomials of degree n.

Original entry on oeis.org

1, 2, 6, 10, 24, 38, 78, 118, 224, 330, 584, 838, 1420, 2002, 3258, 4514, 7134, 9754, 15010, 20266, 30532, 40798, 60280, 79762, 115966, 152170, 217962, 283754, 401250, 518746, 724866, 930986, 1287306, 1643626, 2250538, 2857450, 3878298, 4899146, 6594822

Offset: 0

Franklin T. Adams-Watters, Jul 19 2006

Also the number of types of crystallographic rotations and reflection-rotations in n-dimensional Euclidean space. - Andrey Zabolotskiy, Jul 08 2017

			The six polynomials of degree 2 consist of 3 irreducible cyclotomic polynomials: x^2+1, x^2+x+1 and x^2-x+1 and 3 products of 2 linear cyclotomic polynomials: x^2+2x+1, x^2-1 and x^2-2x+1.
The six plane crystallographic operations are the identity operation, rotations by 2 Pi/k with k = 2,3,4,6, and a reflection.

Boyd, David W.(3-BC); Montgomery, Hugh L.(1-MI), Cyclotomic partitions. In Number theory (Banff, AB, 1988), 7-25. Walter de Gruyter & Co., Berlin, 1990 ISBN:3-11-011723-1, MR1106647. [Asymptotics]

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
Gaëtan Chenevier, The Characteristic Masses of Niemeier Lattices, arXiv:2002.03707 [math.NT], 2020.
Peter Engel, Louis Michel and Marjorie Senechal, Lattice Geometry, 2004 (see section 1.4.3).
Richard P. Stanley, Some enumerative applications of cyclotomic polynomials, preprint, 2024-2025. See p. 13.
D. Weigel, R. Veysseyre, T. Phan, J. M. Effantin, and Y. Billiet, Crystallography, geometry and physics in higher dimensions. I. Point-symmetry operations, Acta Cryst., A40 (1984), 323-330 (see table 3).

Cf. A014197, A051894, A280611 (variant where repeated roots are not allowed).

A057192 Least m such that 1 + prime(n)*2^m is a prime, or -1 if no such m exists.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 6, 1, 1, 8, 2, 1, 2, 583, 1, 5, 4, 2, 3, 2, 2, 1, 1, 2, 3, 16, 3, 6, 1, 2, 1, 3, 2, 3, 4, 8, 2, 7, 1, 1, 4, 1, 2, 15, 2, 20, 8, 11, 6, 1, 1, 36, 1, 279, 29, 3, 4, 2, 1, 30, 1, 2, 9, 4, 7, 4, 4, 3, 10, 21, 1, 12, 2, 14, 6393, 11, 4, 3, 2, 1, 4, 1, 2, 6, 1, 3, 8, 5, 6, 19, 3, 2, 1, 2, 5

Offset: 1

Labos Elemer, Jan 10 2001

sign

Primes p such that p * 2^m + 1 is composite for all m are called Sierpiński numbers. The smallest known prime Sierpiński number is 271129. Currently, 10223 is the smallest prime whose status is unknown.
For 0 < k < a(n), prime(n)*2^k is a nontotient. See A005277. - T. D. Noe, Sep 13 2007
With the discovery of the primality of 10223 * 2^31172165 + 1 on November 6, 2016, we now know that 10223 is not a Sierpiński number. The smallest prime of unknown status is thus now 21181. The smallest confirmed instance of a(n) = -1 is for n = 78557. - Alonso del Arte, Dec 16 2016 [Since we only care about prime Sierpiński numbers in this sequence, 78557 should be replaced by primepi(271129) = 23738. - Jianing Song, Dec 15 2021]
Aguirre conjectured that, for every n > 1, a(n) is even if and only if prime(n) mod 3 = 1 (see the MathStackExchange link below). - Lorenzo Sauras Altuzarra, Feb 12 2021
If prime(n) is not a Fermat prime, then a(n) is also the least m such that prime(n)*2^m is a totient number, or -1 if no such m exists. If prime(n) = 2^2^e + 1 is a Fermat prime, then the least m such that prime(n)*2^m is a totient number is min{2^e, a(n)} if a(n) != -1 or 2^e if a(n) = -1, since 2^2^e * (2^2^e + 1) = phi((2^2^e+1)^2) is a totient number. For example, the least m such that 257*2^m is a totient number is m = 8, rather than a(primepi(257)) = 279; the least m such that 65537*2^m is a totient number is m = 16, rather than a(primepi(65537)) = 287. - Jianing Song, Dec 15 2021

			a(8) = 6 because prime(8) = 19 and the first prime in the sequence 1 + 19 * {2, 4, 8,1 6, 32, 64} = {39, 77, 153, 305, 609, 1217} is 1217 = 1 + 19 * 2^6.

See A046067.

T. D. Noe, Table of n, a(n) for n = 1..1000
Ray Ballinger and Wilfrid Keller, Sierpiński Problem
Timothy Revell, "Crowdsourced prime number could help solve a 50-year-old problem", New Scientist, November 18, 2016.
Tejas R. Rao, Effective primality test for p*2^n+1, p prime, n>1, arXiv:1811.06070 [math.NT], 2018.
Seventeen or Bust, A Distributed Attack on the Sierpinski problem
StackExchange, If p is prime, does there exist a positive integer k such that 2^k*p + 1 is also prime?

Cf. A058887, A058811, A058812, A002202, A014197, A005277, A005384, A005385, A051686.
Cf. A046067 (least k such that (2n - 1) * 2^k + 1 is prime).
a(n) = -1 if and only if n is in A076336.

a := proc(n)
   local m:
   m := 0:
   while not isprime(1+ithprime(n)*2^m) do m := m+1: od:
   m:
end: # Lorenzo Sauras Altuzarra, Feb 12 2021

Table[p = Prime[n]; k = 0; While[Not[PrimeQ[1 + p * 2^k]], k++]; k, {n, 100}] (* T. D. Noe *)

a(n) = my(m=0, p=prime(n)); while (!isprime(1+p*2^m), m++); m; \\ Michel Marcus, Feb 12 2021

Corrected by T. D. Noe, Aug 03 2005

A032446 Number of solutions to phi(k) = 2n.

Original entry on oeis.org

3, 4, 4, 5, 2, 6, 0, 6, 4, 5, 2, 10, 0, 2, 2, 7, 0, 8, 0, 9, 4, 3, 2, 11, 0, 2, 2, 3, 2, 9, 0, 8, 2, 0, 2, 17, 0, 0, 2, 10, 2, 6, 0, 6, 0, 3, 0, 17, 0, 4, 2, 3, 2, 9, 2, 6, 0, 3, 0, 17, 0, 0, 2, 9, 2, 7, 0, 2, 2, 3, 0, 21, 0, 2, 2, 0, 0, 7, 0, 12, 4, 3, 2, 12, 0, 2, 0, 8, 2, 10

Offset: 1

Ursula Gagelmann (gagelmann(AT)altavista.net)

By Carmichael's conjecture, a(n) <> 1 for any n. See A074987. - Thomas Ordowski, Sep 13 2017

a(n) = 0 iff n is a term of A079695. - Bernard Schott, Oct 02 2021

			If n = 8 then phi(x) = 2*8 = 16 is satisfied for only a(8) = 6 values of x, viz. 17, 32, 34, 40, 48, 60.

Albert H. Beiler, Recreations in the Theory of Numbers, The Queen of Mathematics Entertains, Second Edition, Dover Publications, Inc., NY, 1966, page 90.

T. D. Noe, Table of n, a(n) for n = 1..5000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Matteo Caorsi and Sergio Cecotti, Geometric classification of 4d N=2 SCFTs, arXiv:1801.04542 [hep-th], 2018.
Carl Pomerance, Popular values of Euler's function, Mathematica 27 (1980), 84-89.

Bisection of A014197.
Cf. A000010, A005277, A074987, A079695, A085758.
Cf. A006511 (largest k for which A000010(k) = A002202(n)), A057635.

[#EulerPhiInverse( 2*n):n in [1..100]]; // Marius A. Burtea, Sep 08 2019

with(numtheory); [ seq(nops(invphi(2*n)), n=1..90) ];

t = Table[0, {100} ]; Do[a = EulerPhi[n]; If[a < 202, t[[a/2]]++ ], {n, 3, 10^5} ]; t

a(n) = invphiNum(2*n); \\ Amiram Eldar, Nov 15 2024 using Max Alekseyev's invphi.gp

Extended by Robin Trew (trew(AT)hcs.harvard.edu).

A049283 a(n) is the smallest k such that phi(k) = n, where phi is Euler's totient function, or a(n) = 0 if no such k exists.

Original entry on oeis.org

1, 3, 0, 5, 0, 7, 0, 15, 0, 11, 0, 13, 0, 0, 0, 17, 0, 19, 0, 25, 0, 23, 0, 35, 0, 0, 0, 29, 0, 31, 0, 51, 0, 0, 0, 37, 0, 0, 0, 41, 0, 43, 0, 69, 0, 47, 0, 65, 0, 0, 0, 53, 0, 81, 0, 87, 0, 59, 0, 61, 0, 0, 0, 85, 0, 67, 0, 0, 0, 71, 0, 73, 0, 0, 0, 0, 0, 79, 0, 123, 0, 83, 0, 129, 0, 0, 0, 89

Offset: 1

Jud McCranie, Oct 10 2000

nonn

			The smallest k such that phi(k) = 2 is k = 3, so a(2) = 3.

T. D. Noe, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010, A014197, A057635.

Module[{nn=140,ep},ep=Table[{k,EulerPhi[k]},{k,0,nn}];Table[SelectFirst[ep,#[[2]]==n&],{n,nn}]][[;;,1]]/."NotFound"->0 (* Harvey P. Dale, Jul 29 2023 *)

a(n)=if(n>2,for(k=n+1,solve(x=n,2*n^2,x/(exp(Euler)*log(log(x))+3/log(log(x)))-n),if(eulerphi(k)==n,return(k)));0,2*n-1) \\ Charles R Greathouse IV, Nov 28 2012

x=1000;v=vector(x\(exp(Euler)*log(log(x))+3/log(log(x)))); for(n=1,x,t=eulerphi(n); if(t<=#v && !v[t], v[t]=n)); v \\ Charles R Greathouse IV, Nov 28 2012

a(n) = max(0, invphiMin(n)); \\ Amiram Eldar, Nov 15 2024, using Max Alekseyev's invphi.gp

A290077 a(n) = A000010(A005940(1+n)).

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 8, 4, 20, 6, 18, 8, 10, 6, 12, 8, 24, 8, 24, 8, 42, 20, 40, 12, 100, 18, 54, 16, 12, 10, 20, 12, 40, 12, 36, 16, 60, 24, 48, 16, 120, 24, 72, 16, 110, 42, 84, 40, 168, 40, 120, 24, 294, 100, 200, 36, 500, 54, 162, 32, 16, 12, 24, 20, 48, 20, 60, 24, 72, 40, 80, 24, 200, 36, 108, 32, 120, 60, 120

Offset: 0

Antti Karttunen, Jul 19 2017

nonn

Each n occurs A014197(n) times in total in this sequence.

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to binary expansion of n

Cf. A000010, A000040, A005940, A014197, A290076, A323915, A324052, A324054, A364568.

f[n_, i_, x_]:=f[n, i, x]=Which[n==0, x, EvenQ[n], f[n/2, i + 1, x], f[(n - 1)/2, i, x Prime[i]]]; a005940[n_]:=f[n - 1, 1, 1]; Table[EulerPhi[a005940[n + 1]], {n, 0, 100}] (* Indranil Ghosh, Jul 20 2017 *)

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A290077(n) = eulerphi(A005940(1+n));

A290077(n) = { my(p=2,z=1); while(n, if(!(n%2), p=nextprime(1+p), z *= (p-(1==(n%4)))); n>>=1); (z); }; \\ Antti Karttunen, Aug 05 2023

def A290077(n):
    i = 1
    m = 1
    while n > 0:
      if 0==(n%2):
        n = n//2
        i += 1
      else:
        if(1==(n%4)):
          n = (n-1)//4
          m *= sloane.A000040(i)-1
          i += 1
        else:
          n = (n-1)//2
          m *= sloane.A000040(i)
    return m

(define (A290077 n) (A000010 (A005940 (+ 1 n))))

(define (A290077 n) (let loop ((n n) (m 1) (i 1)) (cond ((zero? n) m) ((even? n) (loop (/ n 2) m (+ 1 i))) ((= 1 (modulo n 4)) (loop (/ (- n 1) 4) (* m (- (A000040 i) 1)) (+ 1 i))) (else (loop (/ (- n 1) 2) (* m (A000040 i)) i))))) ;; Requires only an implementation of A000040, see for example under A083221.

a(n) = A000010(A005940(1+n)).

A361967 Number of numbers k such that uphi(k) = n, where uphi is the unitary totient function (A047994).

Original entry on oeis.org

2, 2, 1, 2, 0, 3, 1, 4, 0, 2, 0, 5, 0, 1, 1, 2, 0, 3, 0, 2, 0, 2, 0, 8, 0, 2, 0, 3, 0, 4, 1, 4, 0, 0, 0, 6, 0, 0, 0, 4, 0, 3, 0, 2, 0, 2, 0, 11, 0, 0, 0, 2, 0, 1, 0, 4, 0, 2, 0, 8, 0, 1, 1, 2, 0, 3, 0, 0, 0, 3, 0, 11, 0, 0, 0, 0, 0, 3, 0, 8, 0, 2, 0, 5, 0, 0, 0

Offset: 1

Amiram Eldar, Apr 01 2023

nonn

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

Row lengths of A361966.
The unitary version of A014197.
Cf. A047994, A135347, A327837, A347771 (positions of 0's), A361966, A361968 (indices of records), A361969 (positions of 1's), A361970, A361971 (record values).

a[n_] := Length[invUPhi[n]]; Array[a, 100] (* using the function invUPhi from A361966 *)

a(A347771(n)) = 0.
a(A361969(n)) = 1.
a(A361970(n)) = n.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A327837. - Amiram Eldar, Dec 24 2024

A066412 Number of elements in the set phi_inverse(phi(n)).

Original entry on oeis.org

2, 2, 3, 3, 4, 3, 4, 4, 4, 4, 2, 4, 6, 4, 5, 5, 6, 4, 4, 5, 6, 2, 2, 5, 5, 6, 4, 6, 2, 5, 2, 6, 5, 6, 10, 6, 8, 4, 10, 6, 9, 6, 4, 5, 10, 2, 2, 6, 4, 5, 7, 10, 2, 4, 9, 10, 8, 2, 2, 6, 9, 2, 8, 7, 11, 5, 2, 7, 3, 10, 2, 10, 17, 8, 9, 8, 9, 10, 2, 7, 2, 9, 2, 10, 8, 4, 3, 9, 6, 10, 17, 3, 9, 2, 17, 7

Offset: 1

Vladeta Jovovic, Dec 25 2001

nonn

			invphi(6) = [7, 9, 14, 18], thus a(7) = a(9) = a(14) = a(18) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
Wikipedia, Euler's totient function (see the last paragraph in section "Some values of the function")

Cf. A000010, A001055, A014197, A032447, A036913, A057826, A071181.

Cf. A070305 (positions where coincides with A000005).

Maple
```
nops(invphi(phi(n)));
```

With[{nn = 120}, Function[s, Take[#, nn] &@ Values@ KeySort@ Flatten@ Map[Function[{k, m}, Map[# -> m &, k]] @@ {#, Length@ #} &@ Lookup[s, #] &, Keys@ s]]@ KeySort@ PositionIndex@ Array[EulerPhi, nn^2 + 10]] (* Michael De Vlieger, Jul 18 2017 *)

for(n=1,150,print1(sum(i=1,10*n,if(n-eulerphi(n)-i+eulerphi(i),0,1)),",")) \\ By the original author(s). Note: the upper limit 10*n for the search range is quite ad hoc, and is guaranteed to miss some cases when n is large enough. Cf. Wikipedia-article. - Antti Karttunen, Jul 19 2017

\\ Here is an implementation not using arbitrary limits:
A014197(n, m=1) = { n==1 && return(1+(m<2)); my(p, q); sumdiv(n, d, if( d>=m && isprime(d+1), sum( i=0, valuation(q=n\d, p=d+1), A014197(q\p^i, p))))} \\ M. F. Hasler, Oct 05 2009
A066412(n) = A014197(eulerphi(n)); \\ Antti Karttunen, Jul 19 2017

a(n) = invphiNum(eulerphi(n)); \\ Amiram Eldar, Nov 14 2024, using Max Alekseyev's invphi.gp

;; A naive implementation requiring precomputed A057826:
(define (A066412 n) (if (<= n 2) 2 (let ((ph (A000010 n))) (let loop ((k (A057826 (/ ph 2))) (s 0)) (if (zero? k) s (loop (- k 1) (+ s (if (= ph (A000010 k)) 1 0)))))))) ;; Antti Karttunen, Jul 18 2017

a(n) = Card( k>0 : cototient(k)=cototient(n) ) where cototient(x) = x - phi(x). - Benoit Cloitre, May 09 2002
From Antti Karttunen, Jul 18 2017: (Start)
a(n) = A014197(A000010(n)).
For all n, a(n) <= A071181(n).
(End)

A362181 Number of numbers k such that A323410(k) = n.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 2, 1, 3, 1, 3, 2, 3, 3, 2, 2, 3, 3, 4, 3, 3, 3, 3, 3, 4, 4, 3, 3, 4, 5, 4, 5, 4, 5, 3, 4, 4, 5, 3, 5, 3, 5, 5, 5, 4, 6, 4, 6, 4, 6, 2, 7, 4, 6, 4, 6, 3, 7, 3, 5, 4, 6, 3, 8, 2, 6, 6, 7, 4, 8, 4, 6, 6, 7, 3, 9, 4, 7, 4, 5, 5, 9, 6, 9, 4, 7, 3

Offset: 2

Amiram Eldar, Apr 10 2023

nonn

The offset is 2 since A323410(p) = 1 for all prime powers p (A246655).

a(0) = 1, since there is only one solution, x = 1, to A323410(x) = 0.

Amiram Eldar, Table of n, a(n) for n = 2..10000

Row lengths of A362180.
The unitary version of A063740.
Cf. A246655, A323410, A362182 (positions of 0's), A362183 (indices of records), A362184, A362185 (positions of 1's), A362186.
Similar sequences: A014197, A361967.

ucototient[n_] := n - Times @@ (Power @@@ FactorInteger[n] - 1); ucototient[1] = 0; With[{max = 100}, ucot = Table[ucototient[n], {n, 1, max^2}]; Table[Length[Position[ucot, n]], {n, 2, max}] // Flatten]

a(A362182(n)) = 0.
a(A362185(n)) = 1.
a(A362186(n)) = n.

cp's OEIS Frontend

A048855 Number of integers up to n! relatively prime to n!.

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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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A120963 Number of monic polynomials with integer coefficients of degree n with all roots on the unit circle; number of products of cyclotomic polynomials of degree n.

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A057192 Least m such that 1 + prime(n)*2^m is a prime, or -1 if no such m exists.

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A032446 Number of solutions to phi(k) = 2n.

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A049283 a(n) is the smallest k such that phi(k) = n, where phi is Euler's totient function, or a(n) = 0 if no such k exists.

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