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Search: a109395 -id:a109395
Displaying 1-10 of 22 results found. page 1 2 3
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A331175 Number of values of k, 1 <= k <= n, with A109395(k) = A109395(n), where A109395(n) = n/gcd(n, phi(n)). +20
6
1, 1, 1, 2, 1, 2, 1, 3, 3, 2, 1, 4, 1, 2, 1, 4, 1, 5, 1, 3, 3, 2, 1, 6, 4, 2, 7, 4, 1, 2, 1, 5, 1, 2, 1, 8, 1, 2, 3, 5, 1, 5, 1, 3, 3, 2, 1, 9, 6, 6, 1, 4, 1, 10, 4, 7, 3, 2, 1, 4, 1, 2, 8, 6, 1, 2, 1, 3, 1, 2, 1, 11, 1, 2, 5, 4, 1, 5, 1, 7, 12, 2, 1, 9, 1, 2, 1, 5, 1, 6, 1, 3, 3, 2, 1, 13, 1, 10, 3, 8, 1, 2, 1, 6, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
Ordinal transform of A109395.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..65537
FORMULA
For n >= 1, a(2^n) = n, a(A003277(n)) = 1.
PROG
(PARI)
up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om, invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om, invec[i], (1+pt))); outvec; };
A109395(n) = n/gcd(n, eulerphi(n));
v331175 = ordinal_transform(vector(up_to, n, A109395(n)));
A331175(n) = v331175[n];
CROSSREFS
Cf. A000010, A003277, A009195, A109395.
Cf. also A081373, A330746, A331177.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jan 11 2020
STATUS
approved
A286149 Compound filter: a(n) = T(A046523(n), A109395(n)), where T(n,k) is sequence A000027 used as a pairing function. +20
3
1, 5, 8, 14, 17, 34, 30, 44, 19, 51, 68, 103, 93, 72, 196, 152, 155, 103, 192, 132, 72, 126, 278, 349, 32, 159, 53, 165, 437, 976, 498, 560, 709, 237, 786, 739, 705, 282, 159, 402, 863, 660, 948, 243, 337, 384, 1130, 1273, 49, 132, 1546, 288, 1433, 349, 126, 459, 282, 567, 1772, 2761, 1893, 636, 165, 2144, 2421, 1921, 2280, 390, 2707, 2046, 2558, 2773, 2703 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..10000
MathWorld, Pairing Function
FORMULA
a(n) = (1/2)*(2 + ((A046523(n)+A109395(n))^2) - A046523(n) - 3*A109395(n)).
MATHEMATICA
Table[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 & @@ {Times @@ MapIndexed[ Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]] - Boole[n == 1], Denominator[EulerPhi[n]/n]}, {n, 73}] (* Michael De Vlieger, May 04 2017 *)
PROG
(PARI)
A109395(n) = n/gcd(n, eulerphi(n));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ This function from Charles R Greathouse IV, Aug 17 2011
A286149(n) = (1/2)*(2 + ((A046523(n)+A109395(n))^2) - A046523(n) - 3*A109395(n));
for(n=1, 10000, write("b286149.txt", n, " ", A286149(n)));
(Scheme) (define (A286149 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A109395 n)) 2) (- (A046523 n)) (- (* 3 (A109395 n))) 2)))
(Python)
from sympy import factorint, totient, gcd
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def P(n):
f = factorint(n)
return sorted([f[i] for i in f])
def a046523(n):
x=1
while True:
if P(n) == P(x): return x
else: x+=1
def a(n): return T(a046523(n), n/gcd(n, totient(n))) # Indranil Ghosh, May 05 2017
CROSSREFS
Cf. A000027, A046523, A109395, A285729, A286142, A286143, A286144, A286152, A286154, A286160, A286161, A286162, A286163, A286164.
KEYWORD
nonn
AUTHOR
Antti Karttunen, May 04 2017
STATUS
approved
A308121 Irregular triangle read by rows: T(n,k) = A109395(n)*k-A076512(n)*A038566(n,k). +20
3
0, 1, 1, 2, 1, 1, 1, 2, 3, 4, 2, 1, 1, 2, 3, 4, 5, 6, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 1, 2, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 4, 5, 6, 1, 2, 3, 7, 14, 13, 4, 11, 2, 1, 8 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
Row n has length A000010(n).
Row n > 1 has sum = n*A076512(n)/2.
First value on row(n) = A076511(n).
Last value on row(n) = A076512(n) for n > 1.
For n > 1, A109395(n) = Max(row) + Min(row).
For values x and y on row n > 1 at positions a and b on the row:
x + y = A109395(n), where a = A000010(n) - (b-1).
For n > 2 the penultimate value on row A002110(n) is given by
A038110(n)*A000040(n)-A060753(n).
From Charlie Neder, Jun 05 2019: (Start)
If p is a prime dividing n, then row p*n consists of p copies of row n.
Conjecture: If n is odd, then row 2n can be obtained from row n by interchanging the first and second halves. (End)
LINKS
Jamie Morken, Table of n, a(n) for n = 1..13413 (Rows n = 1..210 of triangle, flattened)
EXAMPLE
The sequence as an irregular triangle:
n/k 1, 2, 3, 4, ...
1: 0
2: 1
3: 1, 2
4: 1, 1
5: 1, 2, 3, 4
6: 2, 1
7: 1, 2, 3, 4, 5, 6
8: 1, 1, 1, 1
9: 1, 2, 1, 2, 1, 2
10: 3, 4, 1, 2
11: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
12: 2, 1, 2, 1
13: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
14: 4, 5, 6, 1, 2, 3
15: 7, 14, 13, 4, 11, 2, 1, 8
...
Row sums: 0, 1, 3, 2, 10, 3, 21, 4, 9, 10, 55, 6, 78, 21, 60.
T(14,5) = A109395(14)*5 - A076512(14)*A038566(14,5) = 7*5 - 3*11 = 2.
T(210,2) = A109395(210)*2 - A076512(210)*A038566(210,2) = 35*2 - 8*11 = -18.
MATHEMATICA
Flatten@ Table[With[{a = n/GCD[n, #], b = Numerator[#/n]}, MapIndexed[a First@ #2 - b #1 &, Flatten@ Position[GCD[Table[Mod[k, n], {k, n - 1}], n], 1] /. {} -> {1}]] &@ EulerPhi@ n, {n, 15}] (* Michael De Vlieger, Jun 06 2019 *)
PROG
(PARI) vtot(n) = select(x->(gcd(n, x)==1), vector(n, k, k));
row(n) = my(q = eulerphi(n)/n, v = vtot(n)); vector(#v, k, denominator(q)*k - numerator(q)*v[k]); \\ Michel Marcus, May 14 2019
CROSSREFS
Cf. A000010, A109395, A038566, A076511, A076512.
KEYWORD
sign,tabf
AUTHOR
Jamie Morken, May 13 2019
STATUS
approved
A002618 a(n) = n*phi(n).
(Formerly M1568 N0611) 		+10
111
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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)
REFERENCES
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).
LINKS
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.
FORMULA
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
EXAMPLE
a(4) = 8 since phi(4) = 2 and 4 * 2 = 8.
a(5) = 20 since phi(5) = 4 and 5 * 4 = 20.
MAPLE
with(numtheory):a:=n->phi(n^2): seq(a(n), n=1..50); # Zerinvary Lajos, Oct 07 2007
MATHEMATICA
Table[n EulerPhi[n], {n, 100}] (* Artur Jasinski, Jan 22 2008 *)
PROG
(MuPAD) numlib::phi(n^2)$ n=1..81 // Zerinvary Lajos, May 13 2008
(Sage) [euler_phi(n^2) for n in range(1, 51)] # Zerinvary Lajos, Jun 06 2009
(Magma) [n*EulerPhi(n): n in [1..150]]; // Vincenzo Librandi, Apr 04 2011
(PARI) a(n)=n*eulerphi(n) \\ Charles R Greathouse IV, Nov 20 2012
(Haskell)
a002618 n = a000010 n * n -- Reinhard Zumkeller, Dec 21 2012
(Python)
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
CROSSREFS
First column of A047916.
Cf. A002619, 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.
KEYWORD
nonn,easy,nice,mult,look
AUTHOR
N. J. A. Sloane
EXTENSIONS
Better description from Labos Elemer, Feb 18 2000
STATUS
approved
A009195 a(n) = gcd(n, phi(n)). +10
60
1, 1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 1, 8, 1, 6, 1, 4, 3, 2, 1, 8, 5, 2, 9, 4, 1, 2, 1, 16, 1, 2, 1, 12, 1, 2, 3, 8, 1, 6, 1, 4, 3, 2, 1, 16, 7, 10, 1, 4, 1, 18, 5, 8, 3, 2, 1, 4, 1, 2, 9, 32, 1, 2, 1, 4, 1, 2, 1, 24, 1, 2, 5, 4, 1, 6, 1, 16, 27, 2, 1, 12, 1, 2, 1, 8, 1, 6, 1, 4, 3, 2, 1, 32, 1, 14, 3, 20 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
The inequality gcd(n, phi(n)) <= 2n exp(-sqrt(log 2 log n)) holds for all squarefree n >= 1 (Erdős, Luca, and Pomerance).
Erdős shows that for almost all n, a(n) ~ log log log log n. - Charles R Greathouse IV, Nov 23 2011
LINKS
T. D. Noe, Table of n, a(n) for n = 1..10000
Paul Erdős, Some asymptotic formulas in number theory, J. Indian Math. Soc. (N.S.) 12 (1948), pp. 75-78.
Paul Erdős, Florian Luca, Carl Pomerance, On the proportion of numbers coprime to a given integer, in Anatomy of Integers, pp. 47-64, J.-M. De Koninck, A. Granville, F. Luca (editors), AMS, 2008.
Joshua Stucky, The distribution of gcd(n,phi(n)), arXiv:2402.13997 [math.NT], 2024.
FORMULA
a(n) = gcd(n, A051953(n)). - Labos Elemer
a(n) = n / A109395(n). - Antti Karttunen, May 04 2017 (corrected also typo in above formula).
MAPLE
a009195 := n -> igcd(i, numtheory[phi](n));
MATHEMATICA
Table[GCD[n, EulerPhi[n]], {n, 100}] (* Harvey P. Dale, Aug 11 2011 *)
PROG
(PARI) a(n)=gcd(n, eulerphi(n)) \\ Charles R Greathouse IV, Nov 23 2011
(Haskell)
a009195 n = n `gcd` a000010 n -- Reinhard Zumkeller, Feb 27 2012
(Python)
def a009195(n):
from math import gcd
phi = lambda x: len([i for i in range(x) if gcd(x, i) == 1])
return gcd(n, phi(n))
# Edward Minnix III, Dec 05 2015
(Magma) [Gcd(n, EulerPhi(n)): n in [1..100]]; // Vincenzo Librandi, Dec 17 2015
CROSSREFS
Cf. A000010, A003277, A050399, A051953, A061303, A109395, A285711.
KEYWORD
nonn,easy,nice
AUTHOR
David W. Wilson
STATUS
approved
A076512 Denominator of cototient(n)/totient(n). +10
20
1, 1, 2, 1, 4, 1, 6, 1, 2, 2, 10, 1, 12, 3, 8, 1, 16, 1, 18, 2, 4, 5, 22, 1, 4, 6, 2, 3, 28, 4, 30, 1, 20, 8, 24, 1, 36, 9, 8, 2, 40, 2, 42, 5, 8, 11, 46, 1, 6, 2, 32, 6, 52, 1, 8, 3, 12, 14, 58, 4, 60, 15, 4, 1, 48, 10, 66, 8, 44, 12, 70, 1, 72, 18, 8, 9, 60, 4, 78, 2, 2, 20, 82, 2, 64, 21 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
a(n)=1 iff n=A007694(k) for some k.
Numerator of phi(n)/n=Prod_{p|n} (1-1/p). - Franz Vrabec, Aug 26 2005
From Wolfdieter Lang, May 12 2011: (Start)
For n>=2, a(n)/A109395(n) = sum(((-1)^r)*sigma_r,r=0..M(n)) with the elementary symmetric functions (polynomials) sigma_r of the indeterminates {1/p_1,...,1/p_M(n)} if n = prod((p_j)^e(j),j=1..M(n)) where M(n)=A001221(n) and sigma_0=1.
This follows by expanding the above given product for phi(n)/n.
The n-th member of this rational sequence 1/2, 2/3, 1/2, 4/5, 1/3, 6/7, 1/2, 2/3, 2/5,... is also (2/n^2)*sum(k,with 1<=k<n and gcd(k,n)=1), n>=2.
Therefore, this scaled sum depends only on the distinct prime factors of n.
See also A023896. Proof via PIE (principle of inclusion and exclusion). (End)
In the sequence of rationals r(n)=eulerphi(n)/n: 1, 1/2, 2/3, 1/2, 4/5, 1/3, 6/7, 1/2, 2/3, 2/5, 10/11, 1/3, ... one can observe that new values are obtained for squarefree indices (A005117); while for a nonsquarefree number n (A013929), r(n) = r(A007947(n)), where A007947(n) is the squarefree kernel of n. - Michel Marcus, Jul 04 2015
LINKS
T. D. Noe, Table of n, a(n) for n=1..1000
FORMULA
a(n) = A000010(n)/A009195(n).
MATHEMATICA
Table[Denominator[(n - EulerPhi[n])/EulerPhi[n]], {n, 80}] (* Alonso del Arte, May 12 2011 *)
PROG
(PARI) vector(80, n, numerator(eulerphi(n)/n)) \\ Michel Marcus, Jul 04 2015
(Magma) [Numerator(EulerPhi(n)/n): n in [1..100]]; // Vincenzo Librandi, Jul 04 2015
CROSSREFS
Cf. A076511 (numerator of cototient(n)/totient(n)), A051953.
Phi(m)/m = k: A000079 \ {1} (k=1/2), A033845 (k=1/3), A000244 \ {1} (k=2/3), A033846 (k=2/5), A000351 \ {1} (k=4/5), A033847 (k=3/7), A033850 (k=4/7), A000420 \ {1} (k=6/7), A033848 (k=5/11), A001020 \ {1} (k=10/11), A288162 (k=6/13), A001022 \ {1} (12/13), A143207 (k=4/15), A033849 (k=8/15), A033851 (k=24/35).
KEYWORD
nonn,frac
AUTHOR
Reinhard Zumkeller, Oct 15 2002
STATUS
approved
A332881 If n = Product (p_j^k_j) then a(n) = denominator of Product (1 + 1/p_j). +10
6
1, 2, 3, 2, 5, 1, 7, 2, 3, 5, 11, 1, 13, 7, 5, 2, 17, 1, 19, 5, 21, 11, 23, 1, 5, 13, 3, 7, 29, 5, 31, 2, 11, 17, 35, 1, 37, 19, 39, 5, 41, 7, 43, 11, 5, 23, 47, 1, 7, 5, 17, 13, 53, 1, 55, 7, 57, 29, 59, 5, 61, 31, 21, 2, 65, 11, 67, 17, 23, 35 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Denominator of sum of reciprocals of squarefree divisors of n.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..20000
FORMULA
Denominators of coefficients in expansion of Sum_{k>=1} mu(k)^2*x^k/(k*(1 - x^k)).
a(n) = denominator of Sum_{d|n} mu(d)^2/d.
a(n) = denominator of psi(n)/n.
a(p) = p, where p is prime.
a(n) = n / A306695(n) = n / gcd(n, A001615(n)). - Antti Karttunen, Nov 15 2021
EXAMPLE
1, 3/2, 4/3, 3/2, 6/5, 2, 8/7, 3/2, 4/3, 9/5, 12/11, 2, 14/13, 12/7, 8/5, 3/2, 18/17, ...
MAPLE
a:= n-> denom(mul(1+1/i[1], i=ifactors(n)[2])):
seq(a(n), n=1..80); # Alois P. Heinz, Feb 28 2020
MATHEMATICA
Table[If[n == 1, 1, Times @@ (1 + 1/#[[1]] & /@ FactorInteger[n])], {n, 1, 70}] // Denominator
Table[Sum[MoebiusMu[d]^2/d, {d, Divisors[n]}], {n, 1, 70}] // Denominator
PROG
(PARI)
A001615(n) = if(1==n, n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A332881(n) = denominator(A001615(n)/n);
CROSSREFS
Cf. A001615, A008683, A017666, A048250, A007947, A109395, A187778 (positions of 1's), A306695, A308443, A308462, A332880 (numerators), A332883.
KEYWORD
nonn,frac
AUTHOR
Ilya Gutkovskiy, Feb 28 2020
STATUS
approved
A076511 Numerator of cototient(n)/totient(n). +10
5
0, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 4, 7, 1, 1, 2, 1, 3, 3, 6, 1, 2, 1, 7, 1, 4, 1, 11, 1, 1, 13, 9, 11, 2, 1, 10, 5, 3, 1, 5, 1, 6, 7, 12, 1, 2, 1, 3, 19, 7, 1, 2, 3, 4, 7, 15, 1, 11, 1, 16, 3, 1, 17, 23, 1, 9, 25, 23, 1, 2, 1, 19, 7, 10, 17, 9, 1, 3, 1, 21, 1, 5, 21, 22, 31, 6, 1, 11, 19, 12, 11 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,6
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..65537
FORMULA
a(n) = A051953(n)/A009195(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A076512(k) = zeta(2)*zeta(3)/zeta(6) - 1 = A082695 - 1 = 0.943596... . Amiram Eldar, Nov 21 2022
MATHEMATICA
Table[Numerator[n/EulerPhi[n] - 1], {n, 1, 100}] (* Amiram Eldar, Nov 21 2022 *)
PROG
(PARI) A076511(n) = numerator((n-eulerphi(n))/eulerphi(n)); \\ Antti Karttunen, Sep 07 2018
CROSSREFS
Cf. A076512 (denominators), A000010, A009195, A051953, A082695, A109395.
KEYWORD
nonn,frac
AUTHOR
Reinhard Zumkeller, Oct 15 2002
STATUS
approved
A259850 Numbers k such that k/phi(k) equals sigma(x)/x for some x<=k. +10
4
1, 3, 8, 9, 14, 15, 16, 21, 26, 27, 28, 32, 40, 45, 50, 52, 56, 63, 64, 75, 80, 81, 98, 100, 104, 112, 128, 130, 135, 144, 147, 160, 162, 182, 189, 192, 196, 200, 208, 216, 224, 225, 243, 250, 255, 256, 260, 288, 310, 320, 324, 338, 364, 372, 375, 384, 392, 400 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
This sequence is motivated by the fact that sigma(n)/n and n/phi(n) are both >= 1.
For the first few terms, we get these ratios: 1, 3/2, 2, 3/2, 7/3, 15/8, 2, ....
The ordered list of distinct values up to a given limit is:
up to 10^1: [1, 3/2, 2];
up to 10^2: [1, 3/2, 7/4, 15/8, 2, 13/6, 7/3, 5/2];
up to 10^3: [1, 3/2, 7/4, 15/8, 31/16, 255/128, 2, 13/6, 7/3, 5/2, 91/36, 31/12, 85/32, 65/24, 35/12, 3, 31/10, 13/4];
up to 10^4: [1, 3/2, 7/4, 15/8, 31/16, 255/128, 2, 13/6, 7/3, 5/2, 91/36, 31/12, 85/32, 65/24, 403/144, 1105/384, 35/12, 635/216, 2555/864, 3, 217/72, 127/42, 73/24, 31/10, 51/16, 13/4, 1651/504, 527/160, 403/120, 221/64, 7/2, 127/36, 217/60];
up to 10^5: [1, 3/2, 7/4, 15/8, 31/16, 255/128, 65535/32768, 2, 33/16, 267/128, 13/6, 7/3, 133/54, 5/2, 91/36, 31/12, 85/32, 21845/8192, 65/24, 11/4, 89/32, 403/144, 1105/384, 35/12, 635/216, 2555/864, 3, 217/72, 127/42, 73/24, 665/216, 595/192, 31/10, 19/6, 51/16, 77/24, 1397/432, 13/4, 1651/504, 527/160, 949/288, 403/120, 221/64, 7/2, 127/36, 511/144, 6851/1920, 217/60, 119/32];
tending towards the intersection of the 2 sets {sigma(n)/n} (A017665/A017666) and {n/phi(n)} (A109395/A076512).
If k is a term, then so are all numbers > k with the same set of prime factors as k. - Robert Israel, Mar 09 2023
LINKS
Robert Israel, Table of n, a(n) for n = 1..1230
EXAMPLE
1/phi(1) = 1/1 = sigma(1)/1, so 1 is in the sequence.
3/phi(3) = 3/2 = sigma(2)/2, so 3 is in the sequence.
8/phi(8) = 2/1 = sigma(6)/6, so 8 is in the sequence.
MAPLE
R:= NULL: count:= 0: V:= {}:
for k from 1 while count < 100 do
V:= V union {numtheory:-sigma(k)/k};
if member(k/numtheory:-phi(k), V) then R:= R, k; count:= count+1 fi;
od:
R; # Robert Israel, Mar 08 2023
PROG
(PARI) lista(nn) = {vs = vector(nn, n, sigma(n)/n); ve = vector(nn, n, n/eulerphi(n)); vr = []; for (n=1, #ve, ven = ve[n]; for (m=1, n, if ((vs[m] == ven), print1(n, ", "); break); ); ); }
CROSSREFS
Cf. A000010, A109395, A076512.
Cf. A000203, A017665, A017666.
Primitive elements: A361363.
KEYWORD
nonn
AUTHOR
Michel Marcus, Jul 07 2015
EXTENSIONS
Name corrected by Michel Marcus, Nov 25 2020
STATUS
approved
A295314 a(n) = sigma(n) / gcd(sigma(n), phi(sigma(n))). +10
4
1, 3, 2, 7, 3, 3, 2, 15, 13, 3, 3, 7, 7, 3, 3, 31, 3, 13, 5, 7, 2, 3, 3, 15, 31, 7, 5, 7, 15, 3, 2, 7, 3, 3, 3, 91, 19, 15, 7, 15, 7, 3, 11, 7, 13, 3, 3, 31, 19, 31, 3, 7, 3, 15, 3, 15, 5, 15, 15, 7, 31, 3, 13, 127, 7, 3, 17, 7, 3, 3, 3, 65, 37, 19, 31, 35, 3, 7, 5, 31, 11, 7, 7, 7, 3, 33, 15, 15, 15, 13, 7, 7, 2, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..16384
FORMULA
a(n) = A000203(n) / A295313(n) = A109395(A000203(n)).
PROG
(PARI) a(n) = my(sn = sigma(n)); sn/gcd(sn, eulerphi(sn)); \\ Michel Marcus, Nov 23 2017
CROSSREFS
Cf. A000010, A000203, A009195, A109395, A295313, A295315.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Nov 22 2017
STATUS
approved
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