A082473 -id:A082473 - cp's OEIS frontend

A194507 a(n) = y is the unique solution to y*phi(y) = A082473(n).

1, 2, 3, 4, 6, 5, 8, 10, 7, 12, 9, 14, 18, 11, 15, 16, 13, 20, 24, 22, 30, 21, 17, 26, 28, 19, 36, 27, 25, 42, 23, 32, 34, 40, 33, 38, 48, 29, 35, 44, 31, 39, 60, 54, 50, 46, 45, 52, 66, 37, 56, 58, 51, 41, 70, 72, 43, 62, 78, 84, 64, 57, 49, 90, 47, 68, 55, 63, 80

Offset: 1

Franz Vrabec, Aug 27 2011

nonn

The permutation which rearranges the terms of A002618 into ascending order. - Antti Karttunen, Sep 28 2019

			a(6) = 5 because 5*phi(5) = 20 = A082473(6).

Antti Karttunen, Table of n, a(n) for n = 1..20001
Index entries for sequences that are permutations of the natural numbers

Cf. A002618, A082473, A327173 (inverse permutation).

Nonzero terms in A327172, in the order of appearance.

Block[{nn = 3000, s, t}, s = Array[EulerPhi[#] (Sqrt@ # /. (c_: 1) a_^(b_: 0) :> (c a^b)^2) &, nn]; t = TakeWhile[Union@ s, # <= nn &]; Map[Block[{y = 1}, While[y EulerPhi@ y != #, y++]; y] &, t]] (* Michael De Vlieger, Sep 29 2019, after Bill Gosper at A007913 *)

up_to = 105;
A327172(n) = { fordiv(n,d,if(eulerphi(d)*d == n, return(d))); (0); };
A194507list(up_to) = { my(v=vector(up_to),k=1); for(n=1,oo,if((v[k]=A327172(n))>0,k++); if(k>up_to, return(v))); };
v194507 = A194507list(up_to);
A194507(n) = v194507[n]; \\ Antti Karttunen, Sep 28 2019

From Antti Karttunen, Sep 28 2019: (Start)
a(n) = A327172(A082473(n)).
A002618(a(n)) = A082473(n).
(End)

A002618 a(n) = n*phi(n).

Original entry on oeis.org

1, 2, 6, 8, 20, 12, 42, 32, 54, 40, 110, 48, 156, 84, 120, 128, 272, 108, 342, 160, 252, 220, 506, 192, 500, 312, 486, 336, 812, 240, 930, 512, 660, 544, 840, 432, 1332, 684, 936, 640, 1640, 504, 1806, 880, 1080, 1012, 2162, 768, 2058, 1000

Offset: 1

N. J. A. Sloane

Also Euler phi function of n^2.
For n >= 3, a(n) is also the size of the automorphism group of the dihedral group of order 2n. This automorphism group is isomorphic to the group of transformations x -> ax + b, where a, b and x are integers modulo n and a is coprime to n. Its order is n*phi(n). - Ola Veshta (olaveshta(AT)my-deja.com), Mar 18 2001
Order of metacyclic group of polynomial of degree n. - Artur Jasinski, Jan 22 2008
It appears that this sequence gives the number of permutations of 1, 2, 3, ..., n that are arithmetic progressions modulo n. - John W. Layman, Aug 27 2008
The conjecture by Layman is correct. Obviously any such permutation must have an increment that is prime to n, and almost as obvious that any such increment will work, with any starting value; hence phi(n) * n total. - Franklin T. Adams-Watters, Jun 09 2009
Consider the numbers from 1 to n^2 written line by line as an n X n square: a(n) = number of numbers that are coprime to all their horizontal and vertical immediate neighbors. - Reinhard Zumkeller, Apr 12 2011
n -> a(n) is injective: a(m) = a(n) implies m = n. - Franz Vrabec, Dec 12 2012 (See Mathematics Stack Exchange link for a proof.)
a(p) = p*(p-1) a pronic number, see A036689 and A002378. - Fred Daniel Kline, Mar 30 2015
Conjecture: All the rational numbers Sum_{i=j..k} 1/a(i) with 0 < min{2,k} <= j <= k have pairwise distinct fractional parts. - Zhi-Wei Sun, Sep 24 2015
From Jianing Song, Aug 25 2023: (Start)
a(n) is the order of the holomorph (see the Wikipedia link) of the cyclic group of order n. Note that Hol(C_n) and Aut(D_{2n}) are isomorphic unless n = 2, where D_{2n} is the dihedral group of order 2*n. See the Wordpress link.
The odd-indexed terms form a subsequence of A341298: the holomorph of an abelian group of odd order is a complete group. See Theorem 3.2, Page 618 of the W. Peremans link. (End)

			a(4) = 8 since phi(4) = 2 and 4 * 2 = 8.
a(5) = 20 since phi(5) = 4 and 5 * 4 = 20.

Peter Giblin, Primes and Programming: An Introduction to Number Theory with Computing. Cambridge: Cambridge University Press (1993) p. 116, Exercise 1.10.
J. L. Lagrange, Oeuvres, Vol. III Paris 1869.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Daniel Fischer, answer to Injectivity of the function n times the Euler Totient of n, Mathematics Stack Exchange, October 2013.
Mikhail R. Gabdullin and Vitalii V. Iudelevich, Numbers of the form kf(k), arXiv:2201.09287 [math.NT] (2022).
Dmitry Krachun and Zhi-Wei Sun, Each positive rational number has the form phi(m^2)/phi(n^2), arXiv:2001.03736 [math.HO], 2020. See also The American Mathematical Monthly (2020) Vol. 127, Issue 9, 847-849.
F. Luca and A. O. Munagi, The number of permutations which form arithmetic progressions modulo m, Annals of the Alexandru Ioan Cuza University, 2014, DOI: 10.2478/aicu-2014-0053. [Broken link]
C. L. Mallows and N. J. A. Sloane, Notes on A002618, A002619, etc.
W. Peremans, Completeness of Holomorphs, Nederl. Akad. Wetensch. Indag. Math. Proc. Ser. A, 60. (1957) 608-619.
N. J. A. Sloane, Notes on A002618, A002619, etc.
J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61.
J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61. [Annotated scanned copy. Note that the scanned pages are out of order]
Wikipedia, Holomorph.
Wordpress, Automorphisms of dihedral groups.

First column of A047916.
Cf. A002619, A011755 (partial sums), A047918, A000010, A053650, A053191, A053192, A036689, A058161, A009262, A082473 (same terms, sorted into ascending order), A256545, A327172 (a left inverse), A327173, A065484.
Subsequence of A323333.

a002618 n = a000010 n * n  -- Reinhard Zumkeller, Dec 21 2012

[n*EulerPhi(n): n in [1..150]]; // Vincenzo Librandi, Apr 04 2011

with(numtheory):a:=n->phi(n^2): seq(a(n), n=1..50); # Zerinvary Lajos, Oct 07 2007

Table[n EulerPhi[n], {n, 100}] (* Artur Jasinski, Jan 22 2008 *)

numlib::phi(n^2)$ n=1..81 // Zerinvary Lajos, May 13 2008

a(n)=n*eulerphi(n) \\ Charles R Greathouse IV, Nov 20 2012

from sympy import totient as phi
def a(n): return n*phi(n)
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Mar 16 2022

[euler_phi(n^2) for n in range(1,51)] # Zerinvary Lajos, Jun 06 2009

Multiplicative with a(p^e) = (p-1)*p^(2e-1). - David W. Wilson, Aug 01 2001
Dirichlet g.f.: zeta(s-2)/zeta(s-1). - R. J. Mathar, Feb 09 2011
a(n) = A173557(n) * A102631(n). - R. J. Mathar, Mar 30 2011
From Wolfdieter Lang, May 12 2011: (Start)
a(n)/2 = A023896(n), n >= 2.
a(n)/2 = (1/n) * Sum_{k=1..n-1, gcd(k,n)=1} k, n >= 2 (see A023896 and A076512/A109395). (End)
a(n) = lcm(phi(n^2),n). - Enrique Pérez Herrero, May 11 2012
a(n) = phi(n^2). - Wesley Ivan Hurt, Jun 16 2013
a(n) = A009195(n) * A009262(n). - Michel Marcus, Oct 24 2013
G.f.: -x + 2*Sum_{k>=1} mu(k)*k*x^k/(1 - x^k)^3. - Ilya Gutkovskiy, Jan 03 2017
a(n) = A082473(A327173(n)), A327172(a(n)) = n. -- Antti Karttunen, Sep 29 2019
Sum_{n>=1} 1/a(n) = 2.203856... (A065484). - Amiram Eldar, Sep 30 2019
Define f(x) = #{n <= x: a(n) <= x}. Gabdullin & Iudelevich show that f(x) ~ c*sqrt(x) for c = Product_{p prime} (1 + 1/(p*(p - 1 + sqrt(p^2 - p)))) = 1.3651304521525857... - Charles R Greathouse IV, Mar 16 2022
a(n) = Sum_{d divides n} A001157(d)*A046692(n/d); that is, the Dirichlet convolution of sigma_2(n) and the Dirichlet inverse of sigma_1(n). - Peter Bala, Jan 26 2024

Better description from Labos Elemer, Feb 18 2000

A327172 If there is a divisor d of n such that phi(d)*d = n, then a(n) = d, otherwise a(n) = 0.

Original entry on oeis.org

1, 2, 0, 0, 0, 3, 0, 4, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 10, 0, 7, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 18, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15

Offset: 1

Antti Karttunen, Sep 28 2019

nonn

If such a divisor exists, it is necessarily unique. See Franz Vrabec's Dec 12 2012 comment in A002618.

Each natural number n > 0 occurs exactly once in this sequence, at position A002618(n).

Antti Karttunen, Table of n, a(n) for n = 1..21000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Left inverse of A002618.
Cf. A000010.
Cf. A082473 (the indices of nonzero terms), A194507 (nonzero terms in the order of appearance).

With[{s = EulerPhi /@ Range@ 120}, Table[DivisorSum[n, # &, # s[[#]] == n &], {n, Length@ s}]] (* Michael De Vlieger, Sep 29 2019 *)

A327172(n) = { fordiv(n,d,if(eulerphi(d)*d == n, return(d))); (0); };

a(A002618(n)) = n.

a(A082473(n)) = A194507(n).

A327170 Number of divisors d of n such that A327171(d) (= phi(d)*core(d)) is equal to n.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Antti Karttunen, Sep 28 2019

nonn

From any solution (*) to A327171(d) = d*phi(d) = n, we obtain a solution for core(d')*phi(d') = n by forming a "pumped up" version d' of d, by replacing each exponent e_i in the prime factorization of d = p_1^e_1 * p_2^e_2 * ... * p_k^e_k, with exponent 2*e_i - 1 so that d' = p_1^(2*e_1 - 1) * p_2^(2*e_2 - 1)* ... * p_k^(2*e_k - 1) = A102631(d) = d*A003557(d), and this d' is also a divisor of n, as n = d' * A173557(d). Generally, any product m = p_1^(2*e_1 - x) * p_2^(2*e_2 - y)* ... * p_k^(2*e_k - z), where each x, y, ..., z is either 0 or 1 gives a solution for core(m)*phi(m) = n, thus every nonzero term in this sequence is a power of 2, even though not all such m's might be divisors of n.
  (* by necessity unique, see Franz Vrabec's Dec 12 2012 comment in A002618).
On the other hand, if we have any solution d for core(d)*phi(d) = n, we can find the unique such divisor e of d that e*phi(e) = n by setting e = A019554(d).
Thus, it follows that the nonzero terms in this sequence occur exactly at positions given by A082473.
Records (1, 2, 4, 8, 16, ...) occur at n = 1, 12, 504, 223200, 50097600, ...

			For n = 504 = 2^3 * 3^2 * 7, it has 24 divisors, out of which four divisors: 42 (= 2^1 * 3^1 * 7^1), 84 (= 2^2 * 3^1 * 7^1), 126 (= 2^1 * 3^2 * 7^1), 252 (= 2^2 * 3^2 * 7^1) are such that A007913(d)*A000010(d) = 504, thus a(504) = 4.

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

Cf. A000010, A002618, A003557, A007913, A019554, A082473, A102631, A173557, A194507, A327171.

Cf. also A327153, A327166, A327169.

With[{s = Array[EulerPhi[#] (Sqrt@ # /. (c_: 1) a_^(b_: 0) :> (c a^b)^2) &, 120]}, Table[DivisorSum[n, 1 &, s[[#]] == n &], {n, Length@ s}]] (* Michael De Vlieger, Sep 29 2019, after Bill Gosper at A007913 *)

A327170(n) = sumdiv(n,d,eulerphi(d)*core(d) == n);

a(n) = Sum_{d|n} [A000010(d)*A007913(d) == n], where [ ] is the Iverson bracket.

A327171 a(n) = phi(n) * core(n), where phi is Euler totient function, and core gives the squarefree part of n.

Original entry on oeis.org

1, 2, 6, 2, 20, 12, 42, 8, 6, 40, 110, 12, 156, 84, 120, 8, 272, 12, 342, 40, 252, 220, 506, 48, 20, 312, 54, 84, 812, 240, 930, 32, 660, 544, 840, 12, 1332, 684, 936, 160, 1640, 504, 1806, 220, 120, 1012, 2162, 48, 42, 40, 1632, 312, 2756, 108, 2200, 336, 2052, 1624, 3422, 240, 3660, 1860, 252, 32, 3120, 1320

Offset: 1

Antti Karttunen, Sep 28 2019

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 161.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Chantal David and Francesco Pappalardi, Average Frobenius distributions of elliptic curves, International Mathematics Research Notices, Vol. 1999, No. 4 (1999), pp. 165-183, alternative link.

Cf. A082473 (gives the terms in ascending order, with duplicates removed).
Cf. A000010, A007913, A053143, A248003, A327170, A327172.
Cf. also A002618, A062355.

[EulerPhi(n)*Squarefree(n): n in [1..100]]; // G. C. Greubel, Jul 13 2024

Array[EulerPhi[#] (Sqrt@ # /. (c_: 1) a_^(b_: 0) :> (c a^b)^2) &, 66] (* Michael De Vlieger, Sep 29 2019, after Bill Gosper at A007913 *)

PARI
```
A327171(n) = eulerphi(n)*core(n);
```

A327171(n) = { my(f=factor(n)); prod (i=1, #f~, (f[i, 1]-1)*(f[i, 1]^(-1 + f[i, 2] + (f[i, 2]%2)))); };

from sympy.ntheory.factor_ import totient, core
def A327171(n):
    return totient(n)*core(n) # Chai Wah Wu, Sep 29 2019

[euler_phi(n)*squarefree_part(n) for n in range(1,101)] # G. C. Greubel, Jul 13 2024

a(n) = A000010(n) * A007913(n).
Multiplicative with a(p^k) = (p-1) * p^((k-1)+(k mod 2)).
Sum_{n>=1} 1/a(n) = (Pi^2/6) * Product_{p prime} (1 + (p+1)/(p^2*(p-1))) = 3.96555686901754604330... - Amiram Eldar, Oct 16 2020
Sum_{k=1..n} a(k) ~ c * n^3, where c = (Pi^2/45) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4) = 0.1500809164... . - Amiram Eldar, Dec 05 2022
a(n) = A000010(A053143(n)). - Amiram Eldar, Sep 15 2023

cp's OEIS Frontend

A194507 a(n) = y is the unique solution to y*phi(y) = A082473(n).

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A002618 a(n) = n*phi(n).

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A327172 If there is a divisor d of n such that phi(d)*d = n, then a(n) = d, otherwise a(n) = 0.

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A327170 Number of divisors d of n such that A327171(d) (= phi(d)*core(d)) is equal to n.

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A327171 a(n) = phi(n) * core(n), where phi is Euler totient function, and core gives the squarefree part of n.

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A327173 Inverse permutation to A194507.

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