A293391 -id:A293391 - cp's OEIS frontend

A011257 Numbers k such that the geometric mean of phi(k) and sigma(k) is an integer.

1, 14, 30, 51, 105, 170, 194, 248, 264, 364, 405, 418, 477, 595, 679, 714, 760, 780, 1023, 1455, 1463, 1485, 1496, 1512, 1524, 1674, 1715, 1731, 1796, 1804, 2058, 2080, 2651, 2754, 2945, 3080, 3135, 3192, 3410, 3534, 3567, 3596, 3828, 3956, 4064, 4381, 4420

Offset: 1

N. J. A. Sloane

nonn

For these terms the arithmetic mean is also an integer. It is conjectured that sigma(k) for these numbers is never odd. See also A065146, A028982, A028983. - Labos Elemer, Oct 18 2001
If p > 2 and 2^p - 1 is prime (a Mersenne prime) then m = 2^(p-2)*(2^p-1) is in the sequence because phi(m) = 2^(p-2)*(2^(p-1)-1); sigma(m) = (2^(p-1)-1)*2^p hence sqrt(phi(m)*sigma(m)) = 2^(p-1)*(2^(p-1)-1) is an integer. So for j > 1, 2^(A000043(j)-2)*2^(A000043(j)-1) is in the sequence. - Farideh Firoozbakht, Nov 27 2005
From a(2633) = 6931232 on, it is no longer true (as was once conjectured) that a(n) > n^2. - M. F. Hasler, Feb 07 2009
It follows from Theorems 1 and 2 in Broughan-Ford-Luca that a(n) << n^(24+e) for all e > 0. - Charles R Greathouse IV, May 09 2013

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 51, p. 19, Ellipses, Paris 2008.
Zhang Ming-Zhi (typescript submitted to Unsolved Problems section of Monthly, 96-01-10).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 2000 terms from M. F. Hasler)
K. Broughan, K. Ford, and F. Luca, On square values of the product of the Euler totient function and sum of divisors function, Colloquium Mathematicum, (to appear).
Tristan Freiberg, Products of shifted primes simultaneously taking perfect power values, Journal of the Australian Mathematical Society 92:2 (2012), pp. 145-154. arXiv:1008.1978 [math.NT], 2010.
Richard K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359-360.
Luis Elesban Santos Cruz and Florian Luca, Power values of the product of the Euler function and the sum of divisors function, involve, Vol. 8 (2015), No. 5, 745-748.

Cf. A000043, A000668.

Cf. A293391 (sigma(m)/phi(m) is a perfect square), A327624 (this sequence \ A293391).

[k:k in [1..4500]| IsPower(EulerPhi(k)*DivisorSigma(1,k),2)]; // Marius A. Burtea, Sep 19 2019

Select[Range[8000], IntegerQ[Sqrt[DivisorSigma[1, #] EulerPhi[#]]] &] (* Carl Najafi, Aug 16 2011 *)

is(n)=issquare(eulerphi(n)*sigma(n)) \\ Charles R Greathouse IV, May 09 2013

A292422 Numbers of the form x = 2^kpq, p,q odd primes, such that sigma(x)/phi(x) = 4.

Original entry on oeis.org

418, 3596, 3956, 5396, 8636, 41656, 56536, 393104, 2072608, 2316448, 6543008, 17434528, 135394432, 217023616, 1264918784, 1490909824, 2710540544, 11444858368, 17669583104, 34797058304, 37698861568, 70572901376, 132968907776, 226965472256, 233356030976, 552070776832, 596357220352, 601188468736

Offset: 1

Jud McCranie, Sep 16 2017

nonn

A subset of A068390 and A020492 (balanced numbers). Conjectured to be infinite by Broughan and Zhou.
From David A. Corneth, Sep 21 2019: (Start)
Exactly 130 terms are of the form 2^35 * p * q.
We have phi and sigma are multiplicative and sigma(2^k) / phi(2^k) = 4 - 2/2^k, and sigma(p)/phi(p) = 1 + 2 / (p-1).
So we need (4 - 2/2^k) * (1 + 2 / (p-1)) <= 4 which gives a lower bound on p depending on k; p > nextprime(4*2^k).
We can then, given k and p, solve for q. Without loss of generality, p < q. Then search over the primes and stop for that value of k when p > q.
This method may be refined using insights from the article and/or given some k, solve the system (1 + 2 / (p-1)) * (1 + 2 / (q - 1)) = (a*m) / (b*m) for p and q where a/b is in lowest terms, m > 0. (End)
Furthermore, p < 8*2^k - 2. - David A. Corneth, Sep 26 2019

			418 = 2*11*19; sigma(418) = 720 = 4*phi(418).

David A. Corneth, Table of n, a(n) for n = 1..221
Kevin A. Broughan and Qizhi Zhou, On the Ratio of the Sum of Divisors and Euler's Totient Function II, Journal of Integer Sequences, vol 17 (2014) article 14.9.2.
David A. Corneth, Some more terms, including terms b-file
David A. Corneth, Some more terms ordered by their exponent of 2, specifying whether it is known that they are all such terms.

Cf. A068390, A020492.

Subsequence of A293391 (sigma(x)/phi(x) square).

is(n) = my(f = factor(n)); #f~ == 3 && f[2, 2] == 1 && f[3, 2] == 1 && f[1,1] == 2 && sigma(f) / eulerphi(f) == 4 \\ David A. Corneth, Sep 21 2019

A327624 Numbers m such that sigma(m)*phi(m) is a square but sigma(m)/phi(m) is not an integer.

Original entry on oeis.org

51, 170, 194, 364, 405, 477, 595, 679, 760, 780, 1023, 1455, 1463, 1496, 1512, 1524, 1674, 1715, 1731, 1796, 1804, 2058, 2080, 2651, 2754, 2945, 3192, 3410, 3534, 3567, 4381, 4420, 5044, 5130, 5670, 5770, 5784, 5797, 5822, 5859, 7600, 8245

Offset: 1

Bernard Schott, Sep 19 2019

nonn

If sigma(m)/phi(m) is a square (m is in A293391) then sigma(m)*phi(m) is also a square (m is in A011257), but the converse is false (see 51 in the Example section). This sequence consists of these counterexamples.

			phi(51) = 32 and sigma(51) = 72, phi(51) * sigma(51) = 32 * 72 = 2304 = 48^2, but sigma(51)/phi(51) = 72/32 = 9/4 is not an integer.

Equals A293391 \ A011257.
Cf. A020492 (sigma(m)/phi(m) is an integer).
Cf. A000010 (phi), A000203 (sigma).

[k:k in [1..9000]| not IsIntegral(SumOfDivisors(k)/EulerPhi(k)) and IsSquare(EulerPhi(k)*SumOfDivisors(k)) ]; // Marius A. Burtea, Sep 19 2019

filter:= v -> sigma(v)/phi(v) <> floor(sigma(v)/phi(v)) and issqr(sigma(v)*phi(v)) : select(filter, [$1..50000]);

sQ[n_] := IntegerQ @ Sqrt[n]; aQ[n_] := sQ[(p = EulerPhi[n]) * (s = DivisorSigma[1, n])] && !sQ[s/p]; Select[Range[10^4], aQ] (* Amiram Eldar, Sep 19 2019 *)

isok(m) = my(s=sigma(m), e=eulerphi(m)); issquare(s*e) && (s%e); \\ Michel Marcus, Sep 21 2019

A341939 Numbers m such that phi(m)/tau(m) is a square of an integer where phi is the Euler totient function (A000010) and tau is the number of divisors function (A000005).

Original entry on oeis.org

1, 3, 8, 10, 18, 19, 24, 30, 34, 45, 52, 57, 73, 74, 85, 102, 125, 135, 140, 152, 153, 156, 163, 182, 185, 190, 202, 219, 222, 252, 255, 333, 342, 360, 375, 394, 416, 420, 436, 451, 455, 456, 459, 476, 489, 505, 514, 546, 555, 570, 584, 606, 625, 629, 640, 646, 679, 680, 730

Offset: 1

Bernard Schott, Feb 24 2021

nonn

The first 11 terms of this sequence are also the first 11 terms of A341938: m such that phi(m)*tau(m) is a square, then, a(12) = 57 while A341938(12) = 54. Indeed, if phi(m)/tau(m) is a perfect square then phi(m)*tau(m) is also a square, but the converse is false. These counterexamples are in A341940, the first one is 54 (last example).
Some subsequences (see examples):
-> The seven terms that satisfy also tau(m) = phi(m) form the subsequence A020488 with phi(m)/tau(m) = 1^2.
-> Primes p of the form 2*k^2 + 1 (A090698) form another subsequence because tau(p) = 2 and phi(p) = p-1 = 2*k^2, so phi(p)/tau(p) = k^2.
-> Cubes p^3 where p is a prime of the form k^2+1 (A002496) form another subset because if p = 2, phi(8)/tau(8)=1, and if p odd, phi(p^3)/tau(p^3) = (k*p/2)^2 with k even.

			phi(30) = 8, tau(30) = 8 so phi(30)/tau(30) = 1^2, and 30 is a term.
phi(45) = 24, tau(45) = 6, so phi(45)/tau(45) = 4 = 2^2, and 85 is a term.
phi(125) = 100, tau(125) = 4, so phi(125)/tau(125) = 25 = 5^2, and 125 is a term.
phi(54) = 18, tau(54) = 8, and phi(54)/tau(54) = 18/8 = 9/4 = (3/2)^2 and 54 is not a term while phi(54)*tau(54) = 12^2.

Intersection of A020491 and A341938.
Similar for: A144695 (sigma(n)/tau(n) perfect square), A293391 (sigma(n)/phi(n) perfect square).
Subsequences: A090698, A020488.
Cf. A069237, A289585, A341940.
Cf. A000005 (phi), A000010(tau).

with(numtheory): filter:= q -> phi(q)/tau(q) = floor(phi(q)/tau(q)) and issqr(phi(q)/tau(q)) : select(filter, [$1..750]);

Select[Range[1000], IntegerQ @ Sqrt[EulerPhi[#]/DivisorSigma[0, #]] &] (* Amiram Eldar, Feb 24 2021 *)

isok(m) = my(x=eulerphi(m)/numdiv(m)); (denominator(x)==1) && issquare(x); \\ Michel Marcus, Feb 24 2021

cp's OEIS Frontend

A011257 Numbers k such that the geometric mean of phi(k) and sigma(k) is an integer.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A292422 Numbers of the form x = 2^k*p*q, p,q odd primes, such that sigma(x)/phi(x) = 4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A327624 Numbers m such that sigma(m)*phi(m) is a square but sigma(m)/phi(m) is not an integer.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

A341939 Numbers m such that phi(m)/tau(m) is a square of an integer where phi is the Euler totient function (A000010) and tau is the number of divisors function (A000005).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

A292422 Numbers of the form x = 2^kpq, p,q odd primes, such that sigma(x)/phi(x) = 4.