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
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.
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Do[If[DivisorSigma[1, m] == 6*EulerPhi[m], Print[m]], {m, 1000000}]
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is(n)=sigma(n)==6*eulerphi(n) \\ Charles R Greathouse IV, May 09 2013
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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
A104901
Numbers n such that sigma(n) = 8*phi(n).
Original entry on oeis.org
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
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.
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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 *)
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is(n)=sigma(n)==8*eulerphi(n) \\ Charles R Greathouse IV, May 09 2013
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
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.
-
Do[If[DivisorSigma[1, m] == 12*EulerPhi[m], Print[m]], {m, 1200000}]
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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
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.
-
Do[If[DivisorSigma[1, m] == 16*EulerPhi[m], Print[m]], {m, 10000000}]
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is(n)=sigma(n)==16*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
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.
-
Do[If[DivisorSigma[1,m]==5*EulerPhi[m],Print[m]],{m,1500000}]
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is(n)=sigma(n)==5*eulerphi(n) \\ Charles R Greathouse IV, May 09 2013
A065819
Numbers k such that 4*phi(k) = 3*sigma(k).
Original entry on oeis.org
7, 209, 10013, 11687, 12857, 17537, 27577, 28067, 700321, 770431, 1321189, 1542281, 1681861, 1963039, 2282641, 2313961, 2664259, 3308041, 3709057, 3859207, 3929761, 4315751, 4380541, 4561381, 5193001, 5331001, 5576519, 5962333, 6561511, 7332919, 10065991, 12133627, 13678613, 14313949, 15263831
Offset: 1
For m=10013, phi(m)=8640, sigma(m)=11520, 34560 = 4*phi = 3*sigma.
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Do[s = 4*EulerPhi[n]-3*DivisorSigma[1, n]; If[Equal[s, 0], Print[n]], {n, 1, 10000000}]
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{ n=0; for (m=1, 10^9, if (4*eulerphi(m) == 3*sigma(m), write("b065819.txt", n++, " ", m); if (n==65, return)) ) } \\ Harry J. Smith, Oct 31 2009
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
- 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.
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Select[Range[10^6], DivisorSigma[1, #] == 10 * EulerPhi[#] &] (* Amiram Eldar, Dec 04 2019 *)
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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
- 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.
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Select[Range[10^6], DivisorSigma[1, #] == 11 * EulerPhi[#] &] (* Amiram Eldar, Dec 04 2019 *)
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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
- 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.
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Select[Range[2*10^6],DivisorSigma[1,#]==13EulerPhi[#]&] (* Harvey P. Dale, Mar 29 2018 *)
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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
- 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.
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Select[Range[10^6], DivisorSigma[1, #] == 14 * EulerPhi[#] &] (* Amiram Eldar, Dec 04 2019 *)
-
for(k=1,2e6, sigma(k) - 14*eulerphi(k) || print1(k", "));
Showing 1-10 of 19 results.
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