A092404 -id:A092404 - cp's OEIS frontend

A067282 Numbers k such that phi(k) + phi(k+1) divides sigma(k) + sigma(k+1).

1, 5, 52, 55, 185, 506, 551, 590, 644, 667, 707, 2285, 2587, 2758, 7551, 10366, 11336, 11564, 11798, 12750, 16616, 16703, 16764, 17383, 18239, 24350, 24415, 26586, 33263, 35541, 40382, 63248, 76247, 76622, 92379, 95069, 97341, 106312, 111388

Offset: 1

Benoit Cloitre, Feb 23 2002

nonn

Presumably the ratio (sigma(n)+sigma(n+1))/(phi(n)+phi(n+1)) can be arbitrarily large. - Labos Elemer, Sep 17 2004

The first term for which the ratio is k for k = 2, 3, ... is 1, 5, 644, 6513584, ... - Amiram Eldar, Mar 02 2020

Amiram Eldar, Table of n, a(n) for n = 1..200

Cf. A000010, A000203, A092403, A092404.

Select[Range[120000], Divisible[DivisorSigma[1, #] + DivisorSigma[1, # + 1], EulerPhi[#] + EulerPhi[# + 1]] &] (* Amiram Eldar, Mar 02 2020 *)
Select[Partition[Table[{n,EulerPhi[n],DivisorSigma[1,n]},{n,111400}],2,1], Divisible[ #[[1,3]]+#[[2,3]],#[[1,2]]+#[[2,2]]]&][[All,1,1]] (* Harvey P. Dale, Apr 25 2020 *)

More terms from Labos Elemer, Sep 17 2004

A197112 Numbers k such that phi(k) = phi(k+1) + phi(k+2).

Original entry on oeis.org

193, 3529, 9337, 27229, 46793, 78181, 90193, 112993, 135013, 437183, 849403, 935219, 1078579, 1283599, 1986973, 2209583, 2341183, 2411173, 2689693, 2744143, 3619069, 3712543, 4738183, 5132983, 6596119, 7829029, 8184713

Offset: 1

Antonio Roldán, Oct 10 2011

nonn

For k less than 4*10^6, k is prime, semiprime, or triprime (3-almost prime).

			112993 is in the sequence, because phi(112993) = 106704, phi(112994) = 48384, phi(112995) = 58320  and 106704 = 48384 + 58320.

Amiram Eldar, Table of n, a(n) for n = 1..289 (calculated using the b-file at A066232)

Cf. A000010, A066232, A092404.

for k from 0 do
        if numtheory[phi](k) = numtheory[phi](k+1)+numtheory[phi](k+2) then
                printf("%d\n",k) ;
        end if;
end do: # R. J. Mathar, Oct 13 2011

Select[Range[10^5], EulerPhi[#] == EulerPhi[# + 1] + EulerPhi[# + 2] &] (* Alonso del Arte, Oct 13 2011 *)
Position[Partition[EulerPhi[Range[82*10^5]],3,1],?(#[[1]]==#[[2]]+#[[3]]&),1,Heads->False]//Flatten (* _Harvey P. Dale, May 10 2022 *)

isok(n) = eulerphi(n) == eulerphi(n+1) + eulerphi(n+2); \\ Michel Marcus, May 15 2016

a(n) = A066232(n)-2. - Donovan Johnson, Oct 14 2011

a(27) from Vincenzo Librandi, Sep 27 2013

A229552 Numbers k such that phi(k) = phi(k+2) - phi(k+1).

Original entry on oeis.org

1, 3, 5, 9, 15, 21, 35, 39, 45, 99, 135, 231, 255, 855, 1035, 1295, 1539, 1599, 2015, 4335, 6525, 9177, 14399, 16095, 30495, 55385, 61131, 62799, 65535, 72579, 77615, 110175, 152649, 179295, 244965, 299649, 603459, 619695, 686735, 1876725, 2841915, 3058209

Offset: 1

Vincenzo Librandi, Sep 27 2013

nonn

Donovan Johnson and Giovanni Resta, Table of n, a(n) for n = 1..324 (terms < 10^13, first 100 terms from Donovan Johnson)

Cf. A092404, A197112, A057000, A000010, A065557.

Select[Range[10^6], EulerPhi[#] == EulerPhi[# + 2] - EulerPhi[# + 1] &]
Position[Partition[EulerPhi[Range[31*10^5]],3,1],?(#[[3]]-#[[2]] == #[[1]]&),1,Heads->False]//Flatten (* _Harvey P. Dale, Nov 29 2017 *)

a(n) = A065557(n) - 2. - Amiram Eldar, Dec 09 2022

A067796 Numbers k such that phi(k) + phi(k+1) = k.

Original entry on oeis.org

14, 20, 44, 140, 254, 272, 284, 344, 464, 764, 1224, 1424, 2064, 3200, 3824, 11648, 21584, 25520, 27644, 38840, 116544, 235584, 603584, 632400, 1677024, 1935344, 2049984, 3763184, 18460224, 74649584, 82448384, 86648000, 97507584, 363939824, 646139264, 4294967294

Offset: 1

Benoit Cloitre, Feb 07 2002

nonn

Cf. A000010, A092404.

Select[Range[10^6], EulerPhi[#] + EulerPhi[# + 1] == # &] (* Amiram Eldar, Apr 23 2022 *)

a(22)-a(33) from Donovan Johnson, Mar 24 2011

a(34)-a(36) from Amiram Eldar, Apr 23 2022

A067797 Numbers k such that phi(k) + phi(k+1) = k+2.

Original entry on oeis.org

4, 6, 8, 50, 74, 110, 170, 174, 890, 902, 1310, 1434, 1826, 2714, 4190, 5030, 6494, 8234, 8474, 10142, 14390, 31670, 32394, 40514, 89990, 95174, 101834, 102554, 135374, 151754, 173942, 206054, 248394, 259694, 288266, 330614, 516242, 650654, 780554, 913994, 941714

Offset: 1

Benoit Cloitre, Feb 07 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..72

Cf. A000010, A092404.

Select[Range[10^6],EulerPhi[#]+EulerPhi[#+1]==#+2&] (* Harvey P. Dale, Apr 04 2019 *)

isok(n) = eulerphi(n) + eulerphi(n+1) == n+2; \\ Michel Marcus, Nov 23 2013

More terms from Michel Marcus, Nov 23 2013

A067799 Numbers k such that phi(k) + phi(k+1) = sigma(k).

Original entry on oeis.org

2, 3, 5, 115, 143, 287, 371, 413, 623, 791, 4067, 10031, 24149, 32335, 39689, 88403, 93443, 107365, 148735, 168833, 212993, 427763, 555947, 650095, 925253, 1017821, 1117457, 1139467, 1229327, 1270801, 1407295, 2184221, 2294509, 2913757, 3258355, 4699981, 5747291

Offset: 1

Benoit Cloitre, Feb 07 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..138

Cf. A000010, A000203, A092404.

Select[Range[10^6], EulerPhi[#] + EulerPhi[# + 1] == DivisorSigma[1, #] &] (* Amiram Eldar, May 21 2022 *)

More terms from Dean Hickerson, Feb 16 2002

More terms from Amiram Eldar, May 21 2022

A076647 Numbers n such that phi(n) + phi(n+1) = sigma(n)/2.

Original entry on oeis.org

56, 456, 476, 828, 846, 1196, 2060, 2780, 4484, 7530, 9174, 11420, 11475, 11865, 13454, 22154, 34916, 35385, 52185, 67308, 105798, 140895, 168872, 190840, 219852, 241892, 244074, 267225, 303834, 317564, 330484, 335440, 460292, 580970, 658515

Offset: 1

Joseph L. Pe, Oct 23 2002

nonn

			phi(56) + phi(57) = 24 + 36 = 60 = 1/2 sigma(56), so 56 is a term of the sequence.

Donovan Johnson, Table of n, a(n) for n = 1..200

Cf. A092404.

with(numtheory): p:=proc(n) if phi(n)+phi(n+1)=sigma(n)/2 then n else fi end: seq(p(n),n=1..2500000); # Emeric Deutsch, Sep 02 2005

Select[Range[10^5], EulerPhi[ # ] + EulerPhi[ # + 1] == (1/2)DivisorSigma[1, # ] &]

isok(n) = eulerphi(n) + eulerphi(n+1) == sigma(n)/2; \\ Michel Marcus, Aug 15 2019

More terms from Emeric Deutsch, Sep 02 2005

A362334 a(n) = A000010(n) + A000010(n+2), where A000010 is the Euler phi-function.

Original entry on oeis.org

3, 3, 6, 4, 10, 6, 12, 8, 16, 8, 22, 10, 20, 14, 24, 14, 34, 14, 30, 18, 34, 18, 42, 20, 38, 24, 46, 20, 58, 24, 50, 32, 44, 28, 60, 30, 60, 34, 64, 28, 82, 32, 66, 42, 70, 38, 88, 36, 74, 44, 84, 42, 92, 42, 76, 52, 94, 44, 118, 46, 96, 62, 84, 52, 114, 52, 110

Offset: 1

Alexandre Herrera, Apr 16 2023

nonn

Conjecture: a(2*n) <= a(2*n-1) and a(2*n) < a(2*n+1).

			For n = 3, phi(3) = 2 and phi(5) = 4, so a(3) = 6.

Cf. A000010, A092404.

a[n_] := Plus @@ EulerPhi[n + {0, 2}]; Array[a, 100] (* Amiram Eldar, Apr 18 2023 *)

a(n) = eulerphi(n) + eulerphi(n+2); \\ Michel Marcus, Apr 17 2023

a(n) = phi(n) + phi(n+2).

cp's OEIS Frontend

A067282 Numbers k such that phi(k) + phi(k+1) divides sigma(k) + sigma(k+1).

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A197112 Numbers k such that phi(k) = phi(k+1) + phi(k+2).

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A229552 Numbers k such that phi(k) = phi(k+2) - phi(k+1).

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A067796 Numbers k such that phi(k) + phi(k+1) = k.

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A067797 Numbers k such that phi(k) + phi(k+1) = k+2.

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A067799 Numbers k such that phi(k) + phi(k+1) = sigma(k).

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A076647 Numbers n such that phi(n) + phi(n+1) = sigma(n)/2.

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A362334 a(n) = A000010(n) + A000010(n+2), where A000010 is the Euler phi-function.

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