Displaying 1-10 of 19 results found.
Numbers n such that sigma(n) = 6*phi(n).
+10
12
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
COMMENTS
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.
EXAMPLE
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).
MATHEMATICA
Do[If[DivisorSigma[1, m] == 6*EulerPhi[m], Print[m]], {m, 1000000}]
PROG
(PARI) 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
Numbers n such that sigma(n) = 8*phi(n).
+10
12
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
COMMENTS
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
EXAMPLE
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).
MATHEMATICA
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 *)
Numbers n such that sigma(n) = 12*phi(n).
+10
10
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
COMMENTS
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.
EXAMPLE
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).
MATHEMATICA
Do[If[DivisorSigma[1, m] == 12*EulerPhi[m], Print[m]], {m, 1200000}]
Numbers n such that sigma(n) = 16*phi(n).
+10
7
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
COMMENTS
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.
EXAMPLE
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).
MATHEMATICA
Do[If[DivisorSigma[1, m] == 16*EulerPhi[m], Print[m]], {m, 10000000}]
Numbers n such that sigma(n) = 5*phi(n).
+10
6
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
COMMENTS
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.
EXAMPLE
sigma(56) = 120 = 5*24 = 5*phi(56) so 56 is in the sequence.
MATHEMATICA
Do[If[DivisorSigma[1, m]==5*EulerPhi[m], Print[m]], {m, 1500000}]
Numbers k such that 4*phi(k) = 3*sigma(k).
+10
5
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
EXAMPLE
For m=10013, phi(m)=8640, sigma(m)=11520, 34560 = 4*phi = 3*sigma.
MATHEMATICA
Do[s = 4*EulerPhi[n]-3*DivisorSigma[1, n]; If[Equal[s, 0], Print[n]], {n, 1, 10000000}]
PROG
(PARI) { 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
Numbers n such that sigma(n) = 10*phi(n) (where sigma= A000203, phi= A000010).
+10
5
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
COMMENTS
If n is in this sequence, then for any prime p not dividing n, sigma(np) - 10*phi(np) = 2*sigma(n).
MATHEMATICA
Select[Range[10^6], DivisorSigma[1, #] == 10 * EulerPhi[#] &] (* Amiram Eldar, Dec 04 2019 *)
PROG
(PARI) for(k=1, 10^6, sigma(k) - 10*eulerphi(k) || print1(k", "));
CROSSREFS
Cf. A062699, A068391, A074400, A068390, A136547, A104900, A136540, A104901, A163667, A171257, A104902, A171258, A171259, A171260, A104903.
Numbers n such that sigma(n) = 11*phi(n) (where sigma= A000203, phi= A000010).
+10
5
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
MATHEMATICA
Select[Range[10^6], DivisorSigma[1, #] == 11 * EulerPhi[#] &] (* Amiram Eldar, Dec 04 2019 *)
PROG
(PARI) for(k=1, 2e6, sigma(k) - 11*eulerphi(k) || print1(k", "));
CROSSREFS
Cf. A062699, A068391, A068390, A136547, A104900, A136540, A104901, A163667, A171256, A104902, A171258, A171259, A171260, A104903.
Numbers n such that sigma(n) = 13*phi(n) (where sigma= A000203, phi= A000010).
+10
5
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
MATHEMATICA
Select[Range[2*10^6], DivisorSigma[1, #]==13EulerPhi[#]&] (* Harvey P. Dale, Mar 29 2018 *)
PROG
(PARI) for(k=1, 2e6, sigma(k) - 13*eulerphi(k) || print1(k", "));
CROSSREFS
Cf. A062699, A068391, A068390, A136547, A104900, A136540, A104901, A163667, A171256, A171257, A104902, A171259, A171260, A104903.
Numbers n such that sigma(n) = 14*phi(n) (where sigma= A000203, phi= A000010).
+10
5
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
MATHEMATICA
Select[Range[10^6], DivisorSigma[1, #] == 14 * EulerPhi[#] &] (* Amiram Eldar, Dec 04 2019 *)
PROG
(PARI) for(k=1, 2e6, sigma(k) - 14*eulerphi(k) || print1(k", "));
CROSSREFS
Cf. A062699, A068391, A068390, A136547, A104900, A136540, A104901, A163667, A171256, A171257, A104902, A171258, A171260, A104903.
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