A033632 -id:A033632 - cp's OEIS frontend

A092587 Numbers k such that sigma(phi(k))-phi(sigma(k)) is nonzero and divisible by phi(k), that is A065395(k)/A000010(k) is a nonzero integer.

2, 18, 21, 99, 133, 151, 175, 183, 350, 366, 449, 450, 477, 532, 581, 645, 702, 843, 1072, 1253, 1346, 1508, 1645, 1833, 2085, 2097, 2150, 2421, 3668, 3950, 4223, 4312, 4453, 5264, 6601, 6853, 7128, 7423, 7622, 7713, 8325, 9028, 9364, 9707, 10820

Offset: 1

Labos Elemer, Mar 01 2004

nonn

			(sigma(phi(x))-phi(sigma(x)))/phi(x) quotient equals -3 for x=450, -2 for x=18, -1 for x=2, 1 for x=21, 2 for x=99, 3 for x=4223.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A033632, A065395, A092584-A092588.

fs[x_] := EulerPhi[DivisorSigma[1, x]] sf[x_] := DivisorSigma[1, EulerPhi[x]] {t=Table[0, {60}], j=1}; Do[s=(sf[n]-fs[n])/EulerPhi[n]; If[ !Equal[s, 0]&&IntegerQ[s], Print[n];t[[j]]=n;j=j+1], {n, 2, 1000000}] t

A132794 Numbers n such that sigma(phi(n)) -phi(n) -1 = phi(sigma(n) -n -1).

Original entry on oeis.org

8, 16, 64, 256, 16384, 262144, 1048576, 4294967296

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Aug 31 2007

Used sigma(n)-n-1, namely the sum of proper divisors minus 1.
a(8) > 10^8. - Michel Marcus, Nov 01 2014
Every 2^(A000043+1) is a term. Proof sketch: Let ch=A048050 and n=2^k, then ch(phi(2^k))=phi(ch(2^k)), ch(2^(k-1))=phi(2^k-2), 2^(k-1)-2=phi(2^(k-1)-1), since phi(prime)=prime-1 the condition is satisfied by every k=A000043+1 or n=2^(A000043+1). See link. - Jon Maiga, Dec 14 2018
Conjecture: a(n)=2^(A000043(n)+1), if true the next terms are: 4294967296, 4611686018427387904, 1237940039285380274899124224... - Jon Maiga, Dec 14 2018
a(9) > 6.5*10^11. - Giovanni Resta, Dec 01 2019

Jon Maiga, Euler Phi, Chowla and Mersenne primes, 2018.

Cf. A000010, A000043, A000203, A001229, A018784, A033632, A048050, A132793.

Filtered([4..1000000],n->Sigma(Phi(n))-Phi(n)-1=Phi(Sigma(n)-n-1)); # Muniru A Asiru, Dec 16 2018

[n: n in [2..30000] | DivisorSigma(1,n) ne n+1 and DivisorSigma(1, EulerPhi(n)) - EulerPhi(n) - 1 eq EulerPhi(DivisorSigma(1,n) - n -1) ]; // G. C. Greubel, Dec 13 2018

with(numtheory); P:=proc(n) local a,i; for i from 1 to n do
a:=phi(sigma(i)-i-1); if a>0 then
if sigma(phi(i))-phi(i)-1=a then print(i);
fi; fi; od; end: P(10^7);

ch[n_]:=DivisorSigma[1,n]-n-1
test[n_]:=ch[n]!=0 && ch[EulerPhi[n]] == EulerPhi[ch[n]]
Flatten[Position[Range[300000], Integer_ ? test]] (* Jon Maiga, Dec 14 2018 *)

isok(n) = ((s=(sigma(n)-n-1)) != 0) && (sigma(eulerphi(n))-eulerphi(n)-1 == eulerphi(s)); \\ Michel Marcus, Nov 01 2014

a(1) corrected and a(6)-a(7) from Michel Marcus, Nov 01 2014

a(8) from Giovanni Resta, Dec 01 2019

A066930 Numbers k such that phi(sigma(k)) divides sigma(phi(k)).

Original entry on oeis.org

1, 7, 9, 29, 71, 97, 109, 121, 139, 142, 175, 183, 194, 215, 225, 242, 244, 261, 278, 311, 344, 349, 355, 430, 497, 509, 516, 533, 556, 571, 605, 622, 631, 647, 673, 709, 729, 791, 817, 859, 911, 923, 1021, 1066, 1112, 1119, 1142, 1207, 1243, 1262, 1277

Offset: 1

Benoit Cloitre, Jan 26 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A000010 (phi), A000203 (sigma), A033632 (subsequence), A062401, A062402.

Select[Range[1300],Divisible[DivisorSigma[1,EulerPhi[#]],EulerPhi[ DivisorSigma[ 1,#]]]&] (* Harvey P. Dale, Dec 15 2021 *)

isok(k) = { sigma(eulerphi(k)) % eulerphi(sigma(k)) == 0 } \\ Harry J. Smith, Apr 08 2010

A092585 Numbers k such that sigma(phi(k))-phi(sigma(k)) is nonzero and is divisible by (k-1), that is A065395(k)/(k-1) = (phi(sigma(k))-sigma(phi(k)))/(k-1) is a nonzero integer.

Original entry on oeis.org

2, 4, 16, 64, 151, 449, 3403, 4096, 4267, 9307, 35905, 65536, 247285, 262144, 17625601, 33126625, 399288961, 649232833, 947278081, 1073741824, 2102485441, 4555788385, 5203567081, 6103058177, 7115716609

Offset: 1

Labos Elemer, Mar 01 2004

			(sigma(phi(x))-phi(sigma(x)))/(x-1) is -1 if x=2,4,16,64,4096,65536,262144 and is 2 if x=151,449,3403, etc.

Cf. A033632, A092584-A092588, A065395.

f[ x_] := EulerPhi[ DivisorSigma[1, x]] - DivisorSigma[1, EulerPhi[x]]; t = {}; Do[ s = f[n]; If[ s != 0 && Mod[ s, n - 1] == 0, Print[n]; AppendTo[t, n]], {n, 2*10^8}]; t

More terms from Robert G. Wilson v, Mar 03 2004

a(17)-a(25) from Donovan Johnson, Mar 04 2013

A227011 Integers m such that phi(sigma(k))/sigma(phi(k)) > phi(sigma(m))/sigma(phi(m)) for all k

Original entry on oeis.org

1, 3, 5, 11, 13, 17, 29, 41, 181, 209, 377, 779, 3239, 4469, 5249, 15539, 43259, 58589, 119279, 169679, 174719, 461369, 692687, 955499, 1258949, 1859129, 1917299, 3925463, 7991693, 8986469, 13244069, 16732169, 30629363, 44137523, 48466987, 64018433, 68787773

Offset: 1

Vladimir Letsko, Oct 09 2013

nonn

These are the indices where the rational function A062401(n)/A062402(n) drops below the minimum set by all earlier ratios.

a(2) to a(9) are primes. However all known terms beginning from a(10) are composite.

			5 is in the sequence because phi(sigma(5))/sigma(phi(5)) = 2/7 and for all k < 5, phi(sigma(k))/sigma(phi(k)) > 2/7.

Donovan Johnson, Table of n, a(n) for n = 1..48 (terms < 10^10)

Cf. A227927, A062401, A062402, A033632, A229238.

A062401 := proc(n)
    numtheory[phi](numtheory[sigma](n))
end proc:
A062402 := proc(n)
    numtheory[sigma](numtheory[phi](n))
end proc:
s := proc(n)
    A062401(n)/A062402(n) ;
end proc:
r := 100000000000000000000000000000 ;
for n from 1 do
    if s(n) < r then
        printf("%d,\n",n) ;
        r := s(n) ;
    end if;
end do:

f(n)=eulerphi(sigma(n=factor(n)))/sigma(eulerphi(n))
is(n)=my(t=f(n)); for(k=1,n-1,if(f(k)<=t, return(0))); 1 \\ Charles R Greathouse IV, Nov 27 2013

a(33)-a(37) from Donovan Johnson, Oct 11 2013

A228567 Primes expressible as sigma(sigma(n)) - sigma(n), in order of their occurrence.

Original entry on oeis.org

3, 7, 17, 31, 31, 41, 23, 73, 127, 73, 89, 127, 463, 523, 241, 523, 157, 241, 523, 463, 211, 257, 131, 983, 379, 1153, 311, 1153, 83, 983, 521, 4339, 4339, 113, 8893, 4339, 4339, 1093, 4339, 769, 2851, 8893, 4339, 1429, 1097, 4339, 1093, 4339, 8893, 4339, 8893

Offset: 1

K. D. Bajpai, Nov 10 2013

nonn

			a(9)= 127: sigma(sigma(93))-sigma(93)= 255-128= 127, which is prime.
a(11)= 89: sigma(sigma(98))-sigma(98)= 260-171= 89, which is prime.

K. D. Bajpai, Table of n, a(n) for n = 1..2200

Cf. A000203 (sigma(n): sum of divisors of n).
Cf. A019279 (superperfect numbers: sigma(sigma(n))=2n).
Cf. A033632 (numbers n: sigma(n)is prime).
Cf. A051027 (sigma(sigma(n))).

with(numtheory):KD := proc() local a; a:= sigma(sigma(n))-sigma(n);if isprime(a) then RETURN (a); fi; end: seq(KD(),n=1..5000);

A073858 Numbers k such that sigma(phi(k)) divides phi(sigma(k)).

Original entry on oeis.org

1, 2, 4, 9, 16, 18, 64, 100, 225, 242, 450, 516, 729, 1458, 3872, 4096, 4624, 13932, 14406, 17672, 18225, 20124, 21780, 28900, 29262, 29616, 36450, 45996, 62500, 65025, 65536, 76832, 92778, 95916, 106092, 106308, 114630, 114930

Offset: 1

Benoit Cloitre, Sep 02 2002

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A033632, A066930.

Select[Range[115000],Divisible[EulerPhi[DivisorSigma[1,#]],DivisorSigma[ 1,EulerPhi[ #]]]&] (* Harvey P. Dale, Jan 31 2021 *)

isok(k) = eulerphi(sigma(k)) % sigma(eulerphi(k))==0 \\ Donovan Johnson, Jul 05 2012

A092589 a(n) = -A065395(2^n).

Original entry on oeis.org

0, 1, 3, 1, 15, 5, 63, 1, 177, 89, 913, -319, 4095, 2393, 10617, 1, 65535, 8897, 262143, -44287, 729537, 543553, 4015777, -1753087, 15622785, 11162969, 46358529, -1452031, 265390977, -2270911, 1073741823, 1, 2668569153, 2862962009, 15344762817, -8238350335, 68103158337, 45811586393

Offset: 0

Labos Elemer, Mar 02 2004

sign

Amiram Eldar, Table of n, a(n) for n = 0..1205

Cf. A000010, A000203, A000225, A033632, A053287, A065395, A092584, A092585, A092586, A092587, A092588.

fs[x_] := EulerPhi[DivisorSigma[1, x]]; sf[x_] := DivisorSigma[1, EulerPhi[x]]; Table[fs[2^w]-sf[2^w], {w, 0, 65}]

a(n) = phi(2^(n+1)-1) - 2^n + 1 = A053287(n+1) - A000225(n). - Amiram Eldar, Jun 09 2024

Offset changed to 0, a(0) prepended and name corrected by Amiram Eldar, Jun 09 2024

A226117 Numbers n such that phi(sigma(tau(n))) = tau(sigma(phi(n))).

Original entry on oeis.org

1, 3, 4, 5, 14, 17, 20, 21, 22, 26, 51, 63, 65, 66, 72, 76, 80, 84, 90, 100, 106, 112, 132, 135, 150, 152, 165, 182, 190, 196, 221, 222, 232, 246, 255, 290, 291, 292, 294, 320, 326, 375, 386, 396, 424, 450, 460, 489, 530, 561, 567, 585, 588, 600, 606, 608, 615

Offset: 1

Paolo P. Lava, May 27 2013

nonn

			For n=23529 we have:
phi(23529)=13200 -> sigma(13200)=46128 -> tau(46128)=30.
tau(23529)=16 -> sigma(16)=31 -> phi(31)=30.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A000005, A000010, A000203, A033632, A067160, A076361, A226118, A226119.

with(numtheory); A226117:=proc(q) local n;
for n from 1 to q do
if phi(sigma(tau(n)))=tau(sigma(phi(n))) then print(n);
fi; od; end: A226117(10^6);

Select[Range[700],EulerPhi[DivisorSigma[1,DivisorSigma[0,#]]] == DivisorSigma[ 0,DivisorSigma[ 1,EulerPhi[ #]]]&] (* Harvey P. Dale, Dec 12 2021 *)

A226118 Numbers n such that sigma(tau(phi(n))) = phi(tau(sigma(n))).

Original entry on oeis.org

1, 2, 136, 160, 170, 204, 240, 282, 716, 745, 1002, 1077, 1465, 1509, 1578, 1868, 2012, 2157, 2346, 2720, 2760, 3608, 3898, 4101, 4461, 4512, 5066, 5322, 5898, 6189, 7080, 7185, 7341, 7628, 7660, 8108, 8517, 8665, 8698, 8709, 8805, 8922, 8940, 9234, 9745, 9962

Offset: 1

Paolo P. Lava, May 27 2013

nonn

			For n=9962 we have:
sigma(9962)=15876 -> tau(15876)=45 -> phi(45)=24.
phi(9962)=4672 -> tau(4672)=14 -> sigma(14)=24.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A000005, A000010, A000203, A033632, A067160, A076361, A226117, A226119.

with(numtheory); A226118:=proc(q) local n;
for n from 1 to q do
if sigma(tau(phi(n)))=phi(tau(sigma(n))) then print(n);
fi; od; end: A226118(10^6);

Select[Range[10000],EulerPhi[DivisorSigma[0,DivisorSigma[1,#]]] == DivisorSigma[ 1, DivisorSigma[ 0, EulerPhi[#]]]&] (* Harvey P. Dale, May 26 2016 *)

cp's OEIS Frontend

A092587 Numbers k such that sigma(phi(k))-phi(sigma(k)) is nonzero and divisible by phi(k), that is A065395(k)/A000010(k) is a nonzero integer.

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A132794 Numbers n such that sigma(phi(n)) -phi(n) -1 = phi(sigma(n) -n -1).

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A066930 Numbers k such that phi(sigma(k)) divides sigma(phi(k)).

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A092585 Numbers k such that sigma(phi(k))-phi(sigma(k)) is nonzero and is divisible by (k-1), that is A065395(k)/(k-1) = (phi(sigma(k))-sigma(phi(k)))/(k-1) is a nonzero integer.

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A227011 Integers m such that phi(sigma(k))/sigma(phi(k)) > phi(sigma(m))/sigma(phi(m)) for all k

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A228567 Primes expressible as sigma(sigma(n)) - sigma(n), in order of their occurrence.

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A073858 Numbers k such that sigma(phi(k)) divides phi(sigma(k)).

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A092589 a(n) = -A065395(2^n).

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