A007694 -id:A007694 - cp's OEIS frontend

A342866 The number of elements in the continued fraction for phi(n)/n, where phi is the Euler totient function (A000010).

1, 2, 3, 2, 3, 2, 3, 2, 3, 3, 3, 2, 3, 3, 4, 2, 3, 2, 3, 3, 4, 3, 3, 2, 3, 3, 3, 3, 3, 4, 3, 2, 6, 3, 5, 2, 3, 3, 6, 3, 3, 3, 3, 3, 4, 3, 3, 2, 3, 3, 6, 3, 3, 2, 5, 3, 6, 3, 3, 4, 3, 3, 4, 2, 7, 4, 3, 3, 6, 4, 3, 2, 3, 3, 4, 3, 6, 3, 3, 3, 3, 3, 3, 3, 4, 3, 6

Offset: 1

Amiram Eldar, Mar 27 2021

			a(2) = 2 since the continued fraction of phi(2)/2 = 1/2 = 0 + 1/2 has 2 elements: {0, 2}.
a(3) = 3 since the continued fraction of phi(3)/3 = 2/3 = 0 + 1/(1 + 1/2) has 3 elements: {0, 1, 2}.
a(15) = 4 since the continued fraction of phi(15)/15 = 8/15 = 0 + 1/(1 + 1/(1 + 1/7)) has 4 elements: {0, 1, 1, 7}.

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

Cf. A000010, A007694, A076512, A109395, A342867.

Cf. A071862 (similar, with sigma(n)/n).

a[n_] := Length @ ContinuedFraction[EulerPhi[n]/n]; Array[a, 100]

a(n) = #contfrac(eulerphi(n)/n); \\ Michel Marcus, Mar 30 2021

a(n) = 2 if and only if n is in A007694.

a(p) = 3 for an odd prime p.

A063872 Let m be the n-th positive integer such that phi(m) is divisible by m - phi(m). Then a(n) = phi(m)/(m - phi(m)).

Original entry on oeis.org

1, 2, 1, 4, 6, 1, 2, 10, 12, 1, 16, 18, 22, 4, 2, 28, 30, 1, 36, 40, 42, 46, 6, 52, 58, 60, 1, 66, 70, 72, 78, 2, 82, 88, 96, 100, 102, 106, 108, 112, 10, 4, 126, 1, 130, 136, 138, 148, 150, 156, 162, 166, 12, 172, 178, 180, 190, 192, 196, 198, 210, 222, 226, 228, 232

Offset: 1

Labos Elemer, Aug 27 2001

m is the n-th prime power larger than 1; i.e., m = A000961(n+1). Proof: If phi(m) is divisible by m-phi(m), then m is divisible by m-phi(m). Let k be the product of the distinct prime factors of m. Then phi(m)/m = phi(k)/k, so k/(k-phi(k)) = m/(m-phi(m)) is an integer. Thus k is divisible by k-phi(k) and k is squarefree. Let k-phi(k) = d and k/(k-phi(k)) = e; note that e>1 and GCD(d,e)=1. Thus d = k - phi(k) = d e - phi(d e) = d e - phi(d) phi(e) so d (e-1) = d e - d = phi(d) phi(e) <= phi(d) (e-1) and d <= phi(d). But this implies that d=1, so phi(k)=k-1 and k is prime. Hence m is a prime power. - Dean Hickerson, Aug 28 2001
For primes, quotient = (p - 1) / 1 = p - 1; for prime powers, p^a, a > 1: quotient = p^(a - 1)(p - 1) / p^(a - 1) = p - 1, so each p - 1 values occur infinitely often: a(n) + 1 = root of n-th prime power with positive exponent, i.e., A025473(n+1). - [Edited by] Daniel Forgues, May 08 2014
"LCM numeral system": a(n+1) is maximum digit for index n, n >= 0; a(-n) is maximum digit for index n, n < 0. - Daniel Forgues, May 03 2014

Harvey P. Dale, Table of n, a(n) for n = 1..1000
OEIS Wiki, LCM numeral system

Cf. A000010, A051953, A007694, A000961, A054740, A049237, A025473.

epd[n_]:=Module[{ep=EulerPhi[n]},If[Divisible[ep,n-ep],ep/(n-ep),Nothing]]; Array[epd,300,2] (* Harvey P. Dale, Dec 27 2020 *)

M(n) = ispower(n, , &n); if (isprime(n), n, 1); \\ A014963
apply(x->x-1, select(isprime, apply(x->M(x+1), [1..260]))) \\ Michel Marcus, Sep 14 2021

a(n) = A025473(n + 1) - 1. - Bill McEachen, Sep 11 2021

A277804 Numbers n such that first digit of n divides n, last digit of n divides n, number of divisors of n divides n and phi(n) divides n, where phi(n) is the Euler totient function.

Original entry on oeis.org

1, 2, 8, 12, 24, 36, 128, 288, 384, 864, 972, 1152, 1944, 3456, 6144, 6912, 13122, 18432, 26244, 31104, 62208, 69984, 209952, 279936, 294912, 497664, 839808, 884736, 1679616, 3538944, 4478976, 13436928, 22674816, 25165824, 31850496, 45349632

Offset: 1

Ilya Gutkovskiy, Nov 01 2016

Numbers n such that A000030(n)|n, A010879(n)|n, A000005(n)|n and A000010(n)|n.

Intersection of A007694, A034709, A033950 and A034837.

			a(5) = 24 because 24/2 = 12, 24/4 = 6, 24 has 8 divisors {1,2,3,4,6,8,12,24}, 24/8 = 3, phi(24) = 8 {1,5,7,11,13,17,19,23} and 24/8 = 3 (all are an integers).

Robert G. Wilson v, Table of n, a(n) for n = 1..43
Index to divisibility sequences

Cf. A000005, A000010, A000030, A007694, A034709, A033950, A034837, A010879.

Select[Range[50000000], Divisible[#1, First[IntegerDigits[#1]]] && Divisible[#1, Last[IntegerDigits[#1]]] && Divisible[#1, DivisorSigma[0, #1]] && Divisible[#1, EulerPhi[#1]] & ]

a(24) - a(36) added by G. C. Greubel, Nov 02 2016

A335327 Numbers k such that iphi(k) divides k, where iphi is an infinitary analog of Euler's phi function (A091732).

Original entry on oeis.org

1, 2, 6, 12, 24, 72, 120, 240, 480, 1440, 2880, 5760, 8640, 17280, 65280, 86400, 120960, 130560, 259200, 391680, 783360, 1566720, 2350080, 4700160, 23500800, 32901120, 47001600, 70502400, 94003200, 470016000, 1410048000, 2820096000, 4294901760, 5640192000, 8460288000

Offset: 1

Amiram Eldar, Jun 01 2020

nonn

			6 is a term since iphi(6) = 2 is a divisor of 6.

Cf. A007694, A091732, A319481.

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], 1])); a[1] = 1; iphi[n_] := Times @@ (Flatten@(f @@@ FactorInteger[n]) - 1);  Select[Range[10^5], Divisible[#,a[#]] &]

A336063 Numbers divisible by the minimal exponent in their prime factorization (A051904).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset: 1

Amiram Eldar, Jul 07 2020

nonn

The asymptotic density of this sequence is 1 (Schinzel and Šalát, 1994).

			4 = 2^2 is a term since A051904(4) = 2 is a divisor of 4.

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, chapter 3, p. 331.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Andrzej Schinzel and Tibor Šalát, Remarks on maximum and minimum exponents in factoring, Mathematica Slovaca, Vol. 44, No. 5 (1994), pp. 505-514.

Cf. A051904, A124184.
A005117 (except for 1) is subsequence.
Similar sequences: A033950, A005349, A006446, A074946, A075592, A007694, A112249, A336064.

h[1] = 0; h[n_] := Min[FactorInteger[n][[;; , 2]]]; Select[Range[2, 100], Divisible[#, h[#]] &]
Select[Range[2,100],Divisible[#,Min[FactorInteger[#][[All,2]]]]&] (* Harvey P. Dale, Aug 31 2020 *)

isok(m) = if (m>1, (m % vecmin(factor(m)[,2])) == 0); \\ Michel Marcus, Jul 08 2020

A373057 Numbers k such that iphi(k) divides k, where iphi is the infinitary Euler phi function (A064380).

Original entry on oeis.org

2, 6, 8, 10, 60, 70, 120, 128, 136, 9822, 18632, 32768, 32896, 36720, 69726, 73662, 73686, 73734, 85962, 86046, 87114, 87198, 87222, 87258, 87294, 87306, 87342, 87366, 87546, 87558, 88014, 88278, 88302, 88338, 88386, 127326, 128046, 128082, 128382, 128406, 128598

Offset: 1

Amiram Eldar, May 21 2024

nonn

Numbers k such that the number of numbers less than k that are infinitarily relatively prime to k is a divisor of k.

			2 is a term since ipghi(2) = 1 divides 2.
6 is a term since ipghi(6) = 6 divides 6.
60 is a term since ipghi(60) = 30 divides 60.

Amiram Eldar, Table of n, a(n) for n = 1..218

Cf. A064380.

Similar sequences: A007694, A097296, A319481, A335327.

infCoprimeQ[n1_, n2_] := Module[{g = GCD[n1, n2]}, If[g == 1, True, AllTrue[ FactorInteger[g][[;; , 1]], BitAnd @@ IntegerExponent[{n1, n2}, #] == 0 &]]]; q[n_] := Divisible[n, Sum[Boole[infCoprimeQ[j, n]], {j, 1, n-1}]]; Select[Range[2, 200], q]

isinfcoprime(n1, n2) = {my(g = gcd(n1, n2), p, e1, e2); if(g == 1, return(1)); p = factor(g)[, 1]; for(i=1, #p, e1 = valuation(n1, p[i]); e2 = valuation(n2, p[i]); if(bitand(e1, e2) > 0, return(0))); 1; }
is(n) = if(n < 2, 0, !(n % sum(j = 1, n-1, isinfcoprime(j, n))));

A054740 Cototient(n)/totient(n) when this is an integer.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Apr 26 2000

nonn

EulerPhi of x divides x (A007694) if and only if EulerPhi divides x-EulerPhi. This quotient is smaller by 1 than A049237.

			x=2592, Phi[2592]=864, Cototient[x]=2592-864=1728 and the quotients are as follows: x/Phi=2592/864=3 or Cototient[x]/Phi[x]=1728/864=2; x always has a form of 2^u*3^w

Sárközy A. and Suranyi J., Number Theory Problem Book (in Hungarian)

A000010, A049237, A007694, A051953.

Select[(#-EulerPhi[#])/EulerPhi[#]&/@Range[300000],IntegerQ] (* Harvey P. Dale, Mar 01 2015 *)

A051953[A007694(n)]/A000010[A007694(n)] = A049237(n)-1

A063755 Squares k which are divisible by phi(k).

Original entry on oeis.org

1, 4, 16, 36, 64, 144, 256, 324, 576, 1024, 1296, 2304, 2916, 4096, 5184, 9216, 11664, 16384, 20736, 26244, 36864, 46656, 65536, 82944, 104976, 147456, 186624, 236196, 262144, 331776, 419904, 589824, 746496, 944784, 1048576, 1327104

Offset: 1

Jason Earls, Aug 14 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..160 from Harry J. Smith)

Cf. A000010, A000290, A049237, A007694. Squares arising in A007694.

Select[ Range[ 1, 2000 ], Mod[ #^2, EulerPhi[ #^2 ] ]==0& ]^2
Select[Range[2000]^2,Divisible[#,EulerPhi[#]]&] (* Harvey P. Dale, Dec 11 2010 *)
Join[{1}, Sort @ Flatten @ Table[2^i*3^j, {i, 2, Log2[m], 2}, {j, 0, Log[3, m/2^i], 2}]] (* Amiram Eldar, Oct 29 2020 *)

j=[]; for(n=1,2000, if(Mod(n^2,eulerphi(n^2))==0,j=concat(j,n^2))); j

{ n=0; for (m=1, 10^9, s=m^2; if (s%eulerphi(s)==0, write("b063755.txt", n++, " ", s); if (n==160, break)) ) } \\ Harry J. Smith, Aug 29 2009

From Amiram Eldar, Oct 29 2020: (Start)
a(n) = A007694(n)^2.
Sum_{n>=1} 1/a(n) = 11/8. (End)

More terms from Dean Hickerson, Aug 15 2001

A064527 Numbers k such that there exists a finite group G of order k such that all entries in its character table are integers.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 24, 32, 36, 48, 54, 64, 72, 96, 108, 120, 128, 144, 162, 192, 200, 216, 240, 256, 288, 324, 384, 400, 432, 480, 486, 512, 576, 648, 720, 768, 800, 864, 960, 972, 1024, 1152, 1200, 1296, 1440, 1458, 1536, 1600, 1728, 1920, 1944

Offset: 1

Tim Brooks (tim_brooks(AT)my-deja.com), Oct 07 2001

The list contains all numbers of the form 2^w*3^u for w > 0, u >= 0. But it also contains 120, 200, 240 and 400. It contains m! for all m because the symmetric groups have integral character tables. By taking direct products, we get all numbers of the form m! * 2^w * 3^u, w > 0, u >= 0. The 200 comes from a semidirect product of an elementary group of order 25 with a quaternion group of order 8, with fixed-point-free action (a Frobenius group). - Derek Holt
From Eric M. Schmidt, Feb 22 2013: (Start)
A group of order k has integral character table iff g^m is conjugate to g for all group elements g and all m coprime to k.
A necessary condition for a group G to have an integral character table is for G/G' to be an elementary Abelian 2-group. Therefore, by the Feit-Thompson theorem, the only odd term in this sequence is 1.
R. Gow proved (see link) that no prime greater than 5 can divide the order of a solvable group with integral character table. (End)
From Jianing Song, Oct 12 2024: (Start)
A finite group whose all characters are rational valued is usually called a Q-group of a rational group, although different authors many define these terms differently.
The unique rational group of order 200 is SmallGroup(200,44) (see Tim Dokchitser's link below). (End)

Tim Dokchitser, Rational groups
Roderick Gow, Groups whose characters are rational-valued, J. Algebra 40 (1976) 280-299.
Groupprops, Finite rational group
Hegedus Pal, Structure of Solvable Rational Groups, Proc. London Math. Soc. (2005) 90 (2): 439-471.

Contains A000142 and A007694.

HasIntegerCharTable := function(G) local cc, ccr, e; cc := ConjugacyClasses(G); ccr := List(cc, Representative); e := Exponent(G); return ForAll([2..e-1], m->(not (IsPrimeInt(m) and GcdInt(m, e)=1)) or ForAll([1..Length(cc)], j->ccr[j]^m in cc[j])); end; A064527 := function(max) local res, i, j; res := [1]; for i in [2, 4..max-(max mod 2)] do if ForAny(res, j->i/j in res) then Add(res, i); continue; fi; for j in [1..NumberSmallGroups(i)] do if HasIntegerCharTable(SmallGroup(i, j)) then Add(res, i); continue; fi; od; od; return res; end; # Eric M. Schmidt, Feb 22 2013

More terms from Derek Holt (mareg(AT)csv.warwick.ac.uk), Oct 07 2001

More terms from Eric M. Schmidt, Feb 22 2013

A066994 Numbers k such that phi(k) divides k*sigma(k).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 24, 26, 27, 30, 32, 35, 36, 39, 40, 42, 48, 54, 55, 56, 60, 63, 64, 70, 72, 78, 80, 84, 88, 96, 98, 104, 105, 108, 110, 114, 116, 120, 125, 126, 128, 135, 140, 144, 147, 150, 155, 156, 160, 162, 165, 168, 189, 190, 192

Offset: 1

Benoit Cloitre, Jan 27 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A000010 (phi), A000203 (sigma), A011775.

Subsequences: A007694, A020492.

Select[Range[200],Divisible[# DivisorSigma[1,#],EulerPhi[#]]&] (* Harvey P. Dale, Aug 23 2019 *)

isok(k) = { (k*sigma(k)) % eulerphi(k) == 0 } \\ Harry J. Smith, Apr 23 2010

Missing term a(7)=10 added by Harry J. Smith, Apr 23 2010

cp's OEIS Frontend

A342866 The number of elements in the continued fraction for phi(n)/n, where phi is the Euler totient function (A000010).

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A063872 Let m be the n-th positive integer such that phi(m) is divisible by m - phi(m). Then a(n) = phi(m)/(m - phi(m)).

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A277804 Numbers n such that first digit of n divides n, last digit of n divides n, number of divisors of n divides n and phi(n) divides n, where phi(n) is the Euler totient function.

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A335327 Numbers k such that iphi(k) divides k, where iphi is an infinitary analog of Euler's phi function (A091732).

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A336063 Numbers divisible by the minimal exponent in their prime factorization (A051904).

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A373057 Numbers k such that iphi(k) divides k, where iphi is the infinitary Euler phi function (A064380).

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A054740 Cototient(n)/totient(n) when this is an integer.

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A063755 Squares k which are divisible by phi(k).

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