A011257 -id:A011257 - cp's OEIS frontend

A164649 Numbers n such that sigma(n)/phi(n) = 36/25.

5797, 10153, 20377, 50953, 383719, 405449, 446039, 486421, 608399, 973709, 1321529, 1521311, 3086369, 3228511, 3451877, 3529813, 3859513, 4552373, 4767721, 5827679, 6194321, 6479599, 6724039, 6927893, 7038241, 7919197, 11696111, 15893773, 16894141, 16924873

Offset: 1

M. F. Hasler, Aug 22 2009

nonn

A subsequence of A011257. See A164646-A164650 for related sequences.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A000010 (=phi), A000203 (=sigma), A068390 (sigma/phi=4), A163667 (sigma/phi=9), A164646-A164650.

for( n=1,1e7, sigma(n)==36/25*eulerphi(n) && print1(n","))

More terms from Sean A. Irvine, May 17 2010

A164650 Numbers n such that sigma(n)/phi(n) = 49/36.

Original entry on oeis.org

679, 10127, 20273, 672203, 971261, 1133639, 1247129, 1336231, 1646743, 1701089, 2369471, 2674969, 2722499, 2989909, 3160079, 3597659, 4545749, 6333503, 7127861, 9357101, 10574629, 20070061, 52928293, 67931137, 74731807, 79940069, 80704813, 93444911, 128155333

Offset: 1

M. F. Hasler, Aug 22 2009

nonn

A subsequence of A011257.
If 7^{k+1}-1 = d*D such that p = 2*7^{k+1}*(d+1)-1 and q = 2*(7^{k+1}+D)-1 are distinct primes, then n = 7^k*p*q is a term of this sequence.
The same theorem holds for sequences of numbers such that sigma/phi=b^2/(b-1)^2 with other primes b (here b=7), cf. A068390, A164646, A164648.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A000010 (=phi), A000203 (=sigma), A068390 (sigma/phi=4), A163667 (sigma/phi=9), A164646-A164649.

for( n=1,1e7, sigma(n)==49/36*eulerphi(n) && print1(n","))

A165630 Numbers n such that sigma(n)/phi(n) = 25/9, where sigma = A000203, phi = A000010.

Original entry on oeis.org

8721, 10179, 21489, 99813, 203721, 228417, 229653, 250705, 268047, 609957, 1150713, 1343277, 2429283, 2835417, 2835807, 2881197, 3150333, 3230469, 3833181, 4679157, 4885569, 5673291, 6082527, 6302529, 6713637, 6819879, 7096329, 9464121, 10313979, 12168651

Offset: 1

Farideh Firoozbakht and M. F. Hasler, Sep 23 2009

nonn

A subsequence of A011257. Contains the product m*n of relatively prime (gcd(m,n)=1) terms (m,n) in A164647 x A164648.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Select[Range[122*10^5],DivisorSigma[1,#]/EulerPhi[#]==25/9&] (* Harvey P. Dale, Jun 20 2021 *)

for( i=1,1e7, sigma(i)/eulerphi(i)==25/9 && print1(i", "))

A327830 Numbers m such that the geometric mean of tau(m) and sigma(m) is an integer.

Original entry on oeis.org

1, 7, 17, 22, 30, 31, 71, 94, 97, 115, 119, 127, 138, 154, 164, 165, 199, 210, 214, 217, 232, 241, 260, 265, 318, 337, 343, 374, 382, 449, 497, 510, 513, 517, 527, 577, 647, 658, 668, 679, 682, 705, 745, 759, 805, 862, 881, 889, 894, 930, 966, 967, 996, 1102, 1125

Offset: 1

Bernard Schott, Sep 27 2019

nonn

The first 20 terms of this sequence are also the first 20 terms of A144695: m such that sigma(m)/tau(m) is a square. Indeed, if sigma(m)/tau(m) is a square then sigma(m)*tau(m) is also a square, but the converse is false. These counterexamples are in A327831; the first one is a(21) = 232.
The primes p of the form 2*k^2 - 1: 7, 17, 31, 71, ... (A066436) form a subsequence because sigma(p) * tau(p) = (2*k)^2.
Another subsequence consists of the terms m such that sigma(m) and tau(m) are both squares; this occurs when m is the product of two distinct primes p*q, p < q where sigma(m) = (p+1)*(q+1) is a square and tau(m) = 4. The first few terms are 22, 94, 115, 119, 214, ... They are in A256152.

			sigma(30) = 72 and tau(30) = 8, sigma(30)*tau(30) = 576 = 24^2, hence 30 is a term.

Cf. A000005 (tau), A000203 (sigma), A064840 (tau*sigma).
Cf. A011257 (similar, with phi(m) and sigma(m)), A144695 (sigma(m)/tau(m) is a square), A327831 (sigma(m) * tau(m) is a square but sigma(m)/tau(m) is not an integer).
Subsequences: A066436, A256152.

[k:k in [1..1150]| IsSquare(#Divisors(k)*DivisorSigma(1,k))]; // Marius A. Burtea, Sep 27 2019

filter:= s -> issqr(sigma(s)*tau(s)) : select(filter, [$1..2500]);

Select[Range[1000], IntegerQ @ Sqrt[DivisorSigma[0, #] * DivisorSigma[1, #]] &] (* Amiram Eldar, Sep 27 2019 *)

isok(m) = issquare(numdiv(m)*sigma(m)); \\ Michel Marcus, Sep 27 2019

A341938 Numbers m such that the geometric mean of tau(m) and phi(m) is 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, 54, 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 A341939: m such that phi(m)/tau(m) is the square of an integer. 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 a(12) = 54.
If k and q are terms and coprimes, then k*q is another term.
Some subsequences (see examples):
-> The seven terms that satisfy tau(m) = phi(m) form the subsequence A020488.
-> 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 tau(p)*phi(p) = (2*k)^2.
-> Cubes p^3 where p is a prime of the form k^2+1 (A002496) form another subset with tau(p^3)*phi(p^3) = (2*k*p)^2.

			phi(18) = tau(18) = 6, so phi(18)*tau(18) = 6^2.
phi(19) = 18, tau(19) = 2, so phi(19)*tau(19) = 36 = 6^2.
phi(34) = 16, tau(34) = 4, so phi(34)*tau(34) = 16*4 = 64 = 8^2.
phi(125) = 100, tau(125) = 4, so phi(125)*tau(125) = 400 = 20^2.

Similar for: A011257 (phi*sigma square), A327830 (sigma*tau square).
Subsequences: A020488, A090698.
Cf. A341939, A341940.
Cf. A000005 (tau), A000010 (phi).

with(numtheory): filter:= n -> issqr(phi(n)*tau(n)) : select(filter, [$1..750]);

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

isok(m) = issquare(numdiv(m)*eulerphi(m)); \\ Michel Marcus, Feb 24 2021

A015705 Geometric mean of phi(n) and sigma(n) is an integer, n odd.

Original entry on oeis.org

1, 51, 105, 405, 477, 595, 679, 1023, 1455, 1463, 1485, 1715, 1731, 2651, 2945, 3135, 3567, 4381, 5797, 5859, 8245, 8255, 8721, 9639, 9809, 10127, 10153, 10179, 10295, 11935, 12369, 17765, 17955, 19125, 19875, 20195, 20213, 20273

Offset: 1

Robert G. Wilson v

nonn

Donovan Johnson, Table of n, a(n) for n = 1..10000
Richard K. Guy, Divisors and desires, Amer. Math. Monthly, 104 (1997), 359-360.

Cf. A011257.

isok(n) = (n%2) && issquare(sigma(n)*eulerphi(n)); \\ Michel Marcus, Oct 02 2017

Offset corrected by Donovan Johnson, Jan 18 2012

A065148 Nonprimes m such that phi(m)*sigma(m) is divisible by m+1.

Original entry on oeis.org

15, 20, 35, 95, 104, 143, 207, 255, 287, 319, 323, 464, 539, 650, 890, 899, 1023, 1034, 1199, 1295, 1349, 1407, 1519, 1763, 1952, 2015, 2204, 2834, 2975, 3599, 4031, 4454, 4607, 5183, 6479, 9215, 9503, 9799, 10403, 11339, 11663, 12095, 12824, 13055

Offset: 1

Labos Elemer, Oct 18 2001

nonn

Every prime p satisfies A000010(p)*A000203(p) == 0 (mod p+1).

			m = 95 is a term since phi(95) = 72, sigma(95) = 120, product = 8640, product/(m+1) = 90.

Amiram Eldar, Table of n, a(n) for n = 1..5000 (terms 1..500 from Harry J. Smith)

Cf. A000010, A000203, A062354, A011257.

Do[s=EulerPhi[n]*DivisorSigma[1, n]; If[IntegerQ[s/(n+1)]&&!PrimeQ[n], Print[n]], {n, 1, 100000}]
Select[Range[14000],!PrimeQ[#]&&Divisible[EulerPhi[#]DivisorSigma[1,#],#+1]&] (* Harvey P. Dale, Jul 08 2017 *)

{ n=0; for (m=1, 10^9, s=eulerphi(m)*sigma(m); if (s%(m+1) == 0 && !isprime(m), write("b065148.txt", n++, " ", m); if (n==500, return)) ) } \\ Harry J. Smith, Oct 12 2009

A000010(m)*A000203(m) == 0 (mod m+1), m is composite.

Offset changed from 0 to 1 by Harry J. Smith, Oct 12 2009

Definition clarified by Harvey P. Dale, Jul 08 2017

A065149 Composite numbers m such that phi(m)*sigma(m) is divisible by m-1.

Original entry on oeis.org

10, 33, 65, 136, 145, 261, 385, 451, 897, 946, 1281, 1441, 1665, 1729, 2241, 2353, 3585, 5185, 6721, 7201, 8380, 8911, 8961, 11521, 11782, 12673, 12801, 17101, 18241, 20737, 25201, 26625, 26677, 26937, 29697, 29953, 30721, 30889, 32896

Offset: 1

Labos Elemer, Oct 18 2001

nonn

			m=136, phi(136)=64, sigma(136)=270, product=17280, quotient=128; for primes the formula holds.

Harry J. Smith, Table of n, a(n) for n = 1..500

Cf. A000010, A000203, A062354, A011257.

Filtered([2..40000],m->Phi(m)*Sigma(m) mod (m-1)=0 and not IsPrime(m)); # Muniru A Asiru, Jun 18 2018

with(numtheory): select(m->modp(phi(m)*sigma(m),m-1)=0 and not isprime(m),[$2..40000]); # Muniru A Asiru, Jun 18 2018

Do[s=EulerPhi[n]*DivisorSigma[1, n]; If[IntegerQ[s/(n-1)]&&!PrimeQ[n], Print[n]], {n, 1, 100000}]

{ n=0; for (m=2, 10^9, s=eulerphi(m)*sigma(m); if (s%(m-1) == 0 && !isprime(m), write("b065149.txt", n++, " ", m); if (n==500, return)) ) } \\ Harry J. Smith, Oct 12 2009

(A000010(m)*A000203(m)) mod (m-1) = 0, m is composite.

Offset changed from 0 to 1 by Harry J. Smith, Oct 12 2009

A065150 Numbers k such that the harmonic mean of phi(k) and sigma(k) is an integer.

Original entry on oeis.org

1, 12, 15, 35, 56, 78, 95, 140, 143, 172, 190, 248, 264, 287, 315, 319, 323, 357, 418, 477, 588, 594, 675, 812, 814, 840, 899, 910, 1045, 1107, 1118, 1131, 1199, 1208, 1254, 1349, 1420, 1425, 1485, 1495, 1558, 1608, 1672, 1763, 2214, 2261, 2318, 2337

Offset: 1

Labos Elemer, Oct 18 2001

nonn

			m = 319, phi(319) = 280, sigma(319) = 360; phi(319)*sigma(319) = 100800, phi(319) + sigma(319) = 640; 1/(harmonic mean) = (640/100800)/2, harmonic mean = 315, arithmetic mean = 320, geometric mean is not an integer.

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

Cf. A000010, A000203, A065146, A011257.

Select[Range[2400], IntegerQ[HarmonicMean @ {EulerPhi[#], DivisorSigma[1, #]}] &] (* Amiram Eldar, Mar 20 2025 *)

{ n=0; for (m=1, 10^9, e=eulerphi(m); s=sigma(m); h=(2*e*s)/(e + s); if (frac(h) == 0, write("b065150.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 13 2009

G^2 mod A = 0, where G^2 = A000010(m)*A000203(m), A = (A000010(m) + A000203(m))/2; harmonic mean = (G^2)/A is an integer.

A165629 Numbers n such that sigma(n)/phi(n) = 25/4, where sigma = A000203, phi = A000010.

Original entry on oeis.org

760, 11020, 18088, 21112, 58206, 65262, 71630, 100280, 123424, 142688, 262276, 303212, 332710, 630344, 679070, 761390, 1265096, 1369120, 1454060, 1454260, 1462552, 1704794, 2185750, 2386664, 2627548, 2783872, 2786056, 2909380, 2927848, 5207680, 5289220

Offset: 1

Farideh Firoozbakht and M. F. Hasler, Sep 23 2009

nonn

A subsequence of A011257. Contains the product m*n of relatively prime (gcd(m,n)=1) terms (m,n) in A068390 x A164648 and in A164646 x A165630.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Select[Range[5300000],4*DivisorSigma[1,#]==25*EulerPhi[#]&] (* Harvey P. Dale, May 09 2012 *)

for( i=1,1e9, sigma(i)*4-25*eulerphi(i) || print1(i", "))

cp's OEIS Frontend

A164649 Numbers n such that sigma(n)/phi(n) = 36/25.

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A164650 Numbers n such that sigma(n)/phi(n) = 49/36.

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A165630 Numbers n such that sigma(n)/phi(n) = 25/9, where sigma = A000203, phi = A000010.

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A327830 Numbers m such that the geometric mean of tau(m) and sigma(m) is an integer.

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A341938 Numbers m such that the geometric mean of tau(m) and phi(m) is an integer where phi is the Euler totient function (A000010) and tau is the number of divisors function (A000005).

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A015705 Geometric mean of phi(n) and sigma(n) is an integer, n odd.

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A065148 Nonprimes m such that phi(m)*sigma(m) is divisible by m+1.

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A065149 Composite numbers m such that phi(m)*sigma(m) is divisible by m-1.

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