A104902 -id:A104902 - cp's OEIS frontend

A104901 Numbers n such that sigma(n) = 8*phi(n).

42, 594, 744, 1254, 7668, 8680, 10788, 11868, 12192, 14630, 15642, 16188, 25908, 28458, 49842, 60078, 70122, 77142, 105222, 124968, 125860, 138460, 142240, 165462, 168402, 169608, 188860, 201924, 242316, 259160, 302260, 553000, 561906, 700910, 726440

Offset: 1

Farideh Firoozbakht, Apr 01 2005

If p>3 and 2^p-1 is prime (a Mersenne prime) then 35*2^(p-2)*(2^p-1) is in the sequence. So 35*2^(A000043-2)*(2^A000043-1) is a subsequence of this sequence.

If p>2 and 2^p-1 is prime (a Mersenne prime) then 3*2^(p-2)*(2^p-1) is in the sequence (the proof is easy). - Farideh Firoozbakht, Dec 23 2007

			p>3, q=2^p-1(q is prime); m=35*2^(p-2)*q so sigma(m)=48*(2^(p-1)-1)*2^p=8*(24*2^(p-3)*(2^p-2))=8*phi(m) hence m is in the sequence.
sigma(553000) = 1497600 = 8*187200 = 8*phi(553000) so 553000 is in the sequence but 553000 is not of the form 35*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, A104902.

Do[If[DivisorSigma[1, m] == 8*EulerPhi[m], Print[m]], {m, 1000000}]
Select[Range[800000],DivisorSigma[1,#]==8*EulerPhi[#]&] (* Harvey P. Dale, Sep 12 2018 *)

is(n)=sigma(n)==8*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", "));

A171260 Numbers n such that sigma(n) = 15*phi(n) (where sigma=A000203, phi=A000010).

Original entry on oeis.org

840, 11880, 12180, 25080, 32130, 67830, 79170, 172260, 282744, 312840, 363660, 569160, 596904, 634410, 696696, 843780, 846090, 959310, 996840, 1119690, 1201560, 1402440, 1542840, 1607340, 1929312, 2104440, 2247210, 2363790, 3309240, 3368040, 3883680

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, A171259, A104903.

Select[Range[10^6], DivisorSigma[1, #] == 15 * EulerPhi[#] &] (* Amiram Eldar, Dec 04 2019 *)

for(k=1,3e6, sigma(k) - 15*eulerphi(k) || print1(k", "));

cp's OEIS Frontend

A104901 Numbers n such that sigma(n) = 8*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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A171257 Numbers n such that sigma(n) = 11*phi(n) (where sigma=A000203, phi=A000010).

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

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

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