A104901 -id:A104901 - cp's OEIS frontend

A062699 Numbers n such that sigma(n) = 2*phi(n).

3, 35, 1045, 24871, 29029, 50065, 58435, 64285, 87685, 137885, 140335, 1390753, 1529983, 1739507, 2011009, 2086903, 3189625, 3281663, 3501605, 3722875, 3830827, 3852155, 6605945, 7711405, 8409305, 9815195, 11413205, 11569805, 13321295, 13932919, 16540205

Offset: 1

Jason Earls, Jul 11 2001

nonn

3 is the only prime term of this sequence. There is no term of the form p^k where p is a prime and k>1. All terms are odd because if n is even then 2*phi(n)=phi(2n)<=nFarideh Firoozbakht, Apr 01 2005, Feb 24 2007

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Jud McCranie, terms 1..50 from Harry J. Smith, terms 51..1000 from Donovan Johnson)
Kevin A. Broughan and Daniel Delbourgo, On the Ratio of the Sum of Divisors and Euler’s Totient Function I, Journal of Integer Sequences, Vol. 16 (2013), Article 13.8.8.
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.

Cf. A068390, A104900, A104901.

Subsequence of A028983 (sigma(k) is even).

Select[Range[10^6], DivisorSigma[1, #] == 2 * EulerPhi[#] &] (* Amiram Eldar, Dec 04 2019 *)

for(n=1,500000, if(sigma(n)==eulerphi(n)*2,print(n)))

n=0; for (m=1, 10^9, if(sigma(m)==2*eulerphi(m), write("b062699.txt", n++, " ", m); if (n==50, break)) ) \\ Harry J. Smith, Aug 09 2009

is(n)=my(f=factor(n)); sigma(f)==2*eulerphi(f) \\ Charles R Greathouse IV, Aug 13 2015

More terms from Labos Elemer, Nov 23 2001

A104900 Numbers n such that sigma(n) = 6*phi(n).

Original entry on oeis.org

6, 70, 616, 1240, 2090, 8932, 17980, 19780, 20320, 26980, 29512, 43180, 49742, 51688, 58058, 79000, 100130, 116870, 128570, 175370, 176715, 201376, 208280, 221536, 275770, 280670, 282680, 302176, 373065, 427924, 435435, 470764, 483616, 618772, 642124

Offset: 1

Farideh Firoozbakht, Apr 01 2005

nonn

If p>2 & 2^p-1 is prime (a Mersenne prime) then 5*2^(p-2)*(2^p-1) is in the sequence. So 5*2^(A000043-2)*(2^A000043-1) is a subsequence of this sequence.

			p>2, q=2^p-1(q is prime); m=5*2^(p-2)*q so sigma(m)=6*(2^(p-1)-1)*2^p=6*phi(m) hence m is in the sequence.
sigma(79000)=187200=6*31200 =6*phi(79000) so 79000 is in the sequence but 79000 is not of the form 5*2^(p-2)*(2^p-1).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Jud McCranie, terms 1..1000 from Donovan Johnson)
Kevin A. Broughan and Daniel Delbourgo, On the Ratio of the Sum of Divisors and Euler’s Totient Function I, Journal of Integer Sequences, Vol. 16 (2013), Article 13.8.8.
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.

Cf. A000043, A062699, A068390, A104901.

Do[If[DivisorSigma[1, m] == 6*EulerPhi[m], Print[m]], {m, 1000000}]

is(n)=sigma(n)==6*eulerphi(n) \\ Charles R Greathouse IV, May 09 2013

v=List(); forfactored(n=6,10^6, if(sigma(n)==6*eulerphi(n), listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, May 09 2017

A104902 Numbers n such that sigma(n) = 12*phi(n).

Original entry on oeis.org

210, 1848, 2970, 3720, 6270, 26796, 38340, 53940, 59340, 60960, 70686, 78210, 80940, 88536, 129540, 142290, 149226, 155064, 174174, 237000, 249210, 300390, 350610, 385710, 429408, 526110, 604128, 624840, 664608, 827310, 828072, 842010, 848040, 906528

Offset: 1

Farideh Firoozbakht, Apr 01 2005

If p>2 and 2^p-1 is prime (a Mersenne prime) then 15*2^(p-2)*(2^p-1) is in the sequence. So 15*2^(A000043-2)*(2^A000043-1) is a subsequence of this sequence.

			p>2, q=2^p-1(q is prime); m=15*2^(p-2)*q so sigma(m)=24*(2^(p-1)-1)*2^p=12*(8*2^(p-3)*(2^p-2))=12*phi(m) hence m is in the sequence.
sigma(237000)=748800=12*62400=12*phi(237000) so 237000 is in the sequence but 237000 is not of the form 15*2^(p-2)*(2^p-1).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Jud McCranie, terms 1..1000 from Donovan Johnson)
Kevin A. Broughan and Daniel Delbourgo, On the Ratio of the Sum of Divisors and Euler’s Totient Function I, Journal of Integer Sequences, Vol. 16 (2013), Article 13.8.8.
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.

Cf. A000043, A062699, A068390, A104900, A104901.

Do[If[DivisorSigma[1, m] == 12*EulerPhi[m], Print[m]], {m, 1200000}]

is(n)=sigma(n)==12*eulerphi(n) \\ Charles R Greathouse IV, May 09 2013

A104903 Numbers n such that sigma(n) = 16*phi(n).

Original entry on oeis.org

20790, 26040, 43890, 268380, 368280, 377580, 415380, 426720, 547470, 566580, 777480, 906780, 996030, 1659000, 1744470, 2102730, 2179320, 2454270, 2699970, 3682770, 4373880, 5053860, 5340060, 5791170, 5874660, 5894070, 5936280, 6035040, 7067340, 8013060

Offset: 1

Farideh Firoozbakht, Apr 01 2005

If p>3 and 2^p-1 is prime (a Mersenne prime) then 105*2^(p-2)*(2^p-1) is in the sequence. So 105*2^(A000043-2)*(2^A000043-1) is a subsequence of this sequence. It seems that 10 divides all terms of this sequence.

			p>2, q=2^p-1(q is prime); m=105*2^(p-2)*q so sigma(m)=192*(2^(p-1)-1)*2^p=16*(48*2^(p-3)*(2^p-2))=16*phi(m) hence m is in the sequence.
sigma(1659000)=5990400=16*374400=16*phi(1659000) so 1659000 is in the sequence but 1659000 is not of the form 105*2^(p-2)*(2^p-1).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Jud McCranie, terms 1..1000 from Donovan Johnson)
Kevin A. Broughan and Daniel Delbourgo, On the Ratio of the Sum of Divisors and Euler’s Totient Function I, Journal of Integer Sequences, Vol. 16 (2013), Article 13.8.8.
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.

Cf. A000043, A062699, A068390, A104900, A104901, A104902.

Do[If[DivisorSigma[1, m] == 16*EulerPhi[m], Print[m]], {m, 10000000}]

is(n)=sigma(n)==16*eulerphi(n) \\ Charles R Greathouse IV, May 09 2013

A136540 Numbers n such that sigma(n) = 7*phi(n).

Original entry on oeis.org

12, 78, 140, 910, 2214, 4180, 4674, 8008, 16120, 25758, 27170, 46816, 54530, 58302, 94240, 99484, 116116, 200260, 233740, 257140, 264160, 350740, 371898, 383656, 479864, 518022, 523218, 551540, 561340, 575598, 616722, 646646, 785118, 965960, 1027000

Offset: 1

Farideh Firoozbakht, Jan 05 2008

If 2^p-1 is a Mersenne prime greater than 3 then m = 65*2^(p-2)*(2^p-1) is in the sequence (the proof is easy).

			sigma(12) = 28 = 7*phi(12) so 12 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Jud McCranie, terms 1..1000 from Donovan Johnson)
Kevin A. Broughan and Daniel Delbourgo, On the Ratio of the Sum of Divisors and Euler’s Totient Function I, Journal of Integer Sequences, Vol. 16 (2013), Article 13.8.8.
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.

Cf. A000043, A000668, A068390, A104900, A104901, A104902, A104903.

with(numtheory): A136540:=n->`if`(sigma(n)=7*phi(n), n, NULL): seq(A136540(n), n=1..10^5); # Wesley Ivan Hurt, Feb 11 2017

Do[If[DivisorSigma[1,n]==7*EulerPhi[n],Print[n]],{n,600000}]
(* Second program *)
Select[Range[10^6], DivisorSigma[1, #] == 7 EulerPhi@ # &] (* Michael De Vlieger, Feb 12 2017 *)

is(n)=sigma(n)==7*eulerphi(n) \\ Charles R Greathouse IV, May 09 2013

A136547 Numbers n such that sigma(n) = 5*phi(n).

Original entry on oeis.org

56, 190, 812, 1672, 4522, 5278, 16065, 24244, 25070, 33915, 39585, 56252, 80104, 93496, 102856, 107156, 140296, 157586, 220616, 224536, 316274, 317205, 365638, 389732, 423045, 479655, 546592, 559845, 596666, 601312, 696514, 731962, 1123605, 1161508, 1181895

Offset: 1

Farideh Firoozbakht, Jan 29 2008, Jan 30 2008

nonn

If p>2 and 2^p-1 is prime (a Mersenne prime) then 377*2^(p-2)*(2^p-1) is in the sequence (the proof is easy). So for n>1 377*2^(A000043(n)-2)*(2^A000043(n)-1) is in the sequence.

			sigma(56) = 120 = 5*24 = 5*phi(56) so 56 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Jud McCranie, terms 1..1000 from Donovan Johnson)
Kevin A. Broughan and Daniel Delbourgo, On the Ratio of the Sum of Divisors and Euler’s Totient Function I, Journal of Integer Sequences, Vol. 16 (2013), Article 13.8.8.
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.

Cf. A000043, A062699, A104900, A104901, A104902.

Do[If[DivisorSigma[1,m]==5*EulerPhi[m],Print[m]],{m,1500000}]

is(n)=sigma(n)==5*eulerphi(n) \\ Charles R Greathouse IV, May 09 2013

A171256 Numbers n such that sigma(n) = 10*phi(n) (where sigma=A000203, phi=A000010).

Original entry on oeis.org

168, 270, 570, 2376, 2436, 5016, 6426, 7110, 13566, 15834, 34452, 58520, 62568, 72732, 75210, 113832, 126882, 168756, 169218, 191862, 199368, 223938, 240312, 280488, 308568, 321468, 420888, 449442, 472758, 661848, 673608, 776736, 848540, 854496, 907236

Offset: 1

M. F. Hasler, Mar 19 2010

nonn

If n is in this sequence, then for any prime p not dividing n, sigma(np) - 10*phi(np) = 2*sigma(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Jud McCranie, terms 1..1000 from Donovan Johnson)
Kevin A. Broughan and Daniel Delbourgo, On the Ratio of the Sum of Divisors and Euler’s Totient Function I, Journal of Integer Sequences, Vol. 16 (2013), Article 13.8.8.
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.

Cf. A062699, A068391, A074400, A068390, A136547, A104900, A136540, A104901, A163667, A171257, A104902, A171258, A171259, A171260, A104903.

Select[Range[10^6], DivisorSigma[1, #] == 10 * EulerPhi[#] &] (* Amiram Eldar, Dec 04 2019 *)

for(k=1,10^6, sigma(k) - 10*eulerphi(k) || print1(k", "));

A171257 Numbers n such that sigma(n) = 11*phi(n) (where sigma=A000203, phi=A000010).

Original entry on oeis.org

2580, 16770, 18630, 28896, 35970, 61404, 66024, 147576, 163944, 215124, 224010, 296184, 399126, 408672, 443394, 464340, 476010, 574308, 856086, 862752, 868428, 931224, 957348, 1004910, 1110186, 1496610, 1721720, 1723290, 1833348, 1971288, 2139852, 2234790

Offset: 1

Farideh Firoozbakht and M. F. Hasler, Mar 19 2010

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Jud McCranie, terms 1..1000 from Donovan Johnson)
Kevin A. Broughan and Daniel Delbourgo, On the Ratio of the Sum of Divisors and Euler’s Totient Function I, Journal of Integer Sequences, Vol. 16 (2013), Article 13.8.8.
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.

Cf. A062699, A068391, A068390, A136547, A104900, A136540, A104901, A163667, A171256, A104902, A171258, A171259, A171260, A104903.

Select[Range[10^6], DivisorSigma[1, #] == 11 * EulerPhi[#] &] (* Amiram Eldar, Dec 04 2019 *)

for(k=1,2e6, sigma(k) - 11*eulerphi(k) || print1(k", "));

A171258 Numbers n such that sigma(n) = 13*phi(n) (where sigma=A000203, phi=A000010).

Original entry on oeis.org

630, 5544, 11160, 18810, 27000, 57000, 80388, 161820, 178020, 182880, 242820, 265608, 388620, 391500, 447678, 465192, 522522, 671760, 690120, 711000, 775170, 826500, 901170, 1051830, 1102290, 1157130, 1418160, 1578330, 1679400, 1812384, 1874520, 1993824

Offset: 1

Farideh Firoozbakht and M. F. Hasler, Mar 19 2010

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Jud McCranie, terms 1..1000 from Donovan Johnson)
Kevin A. Broughan and Daniel Delbourgo, On the Ratio of the Sum of Divisors and Euler’s Totient Function I, Journal of Integer Sequences, Vol. 16 (2013), Article 13.8.8.
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.

Cf. A062699, A068391, A068390, A136547, A104900, A136540, A104901, A163667, A171256, A171257, A104902, A171259, A171260, A104903.

Select[Range[2*10^6],DivisorSigma[1,#]==13EulerPhi[#]&] (* Harvey P. Dale, Mar 29 2018 *)

for(k=1,2e6, sigma(k) - 13*eulerphi(k) || print1(k", "));

A171259 Numbers n such that sigma(n) = 14*phi(n) (where sigma=A000203, phi=A000010).

Original entry on oeis.org

420, 2730, 5940, 12540, 24024, 38610, 48360, 66528, 77490, 81510, 133920, 140448, 141372, 156420, 163590, 282720, 284580, 298452, 348348, 498420, 600780, 681912, 701220, 771420, 792480, 901530, 918918, 1016730, 1052220, 1150968, 1372680, 1439592, 1654620

Offset: 1

Farideh Firoozbakht and M. F. Hasler, Mar 19 2010

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Jud McCranie, terms 1..1000 from Donovan Johnson)
Kevin A. Broughan and Daniel Delbourgo, On the Ratio of the Sum of Divisors and Euler’s Totient Function I, Journal of Integer Sequences, Vol. 16 (2013), Article 13.8.8.
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.

Cf. A062699, A068391, A068390, A136547, A104900, A136540, A104901, A163667, A171256, A171257, A104902, A171258, A171260, A104903.

Select[Range[10^6], DivisorSigma[1, #] == 14 * EulerPhi[#] &] (* Amiram Eldar, Dec 04 2019 *)

for(k=1,2e6, sigma(k) - 14*eulerphi(k) || print1(k", "));

cp's OEIS Frontend

A062699 Numbers n such that sigma(n) = 2*phi(n).

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A104900 Numbers n such that sigma(n) = 6*phi(n).

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A104902 Numbers n such that sigma(n) = 12*phi(n).

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A104903 Numbers n such that sigma(n) = 16*phi(n).

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A136540 Numbers n such that sigma(n) = 7*phi(n).

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A136547 Numbers n such that sigma(n) = 5*phi(n).

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A171256 Numbers n such that sigma(n) = 10*phi(n) (where sigma=A000203, phi=A000010).

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