A065395 -id:A065395 - cp's OEIS frontend

A092588 Numbers k such that sigma(phi(k)) - phi(sigma(k)) is nonzero and divisible by sigma(k), that is A065395(k)/A000203(k) is a nonzero integer.

7, 327, 463, 497, 617, 691, 751, 1207, 1633, 2451, 2643, 3143, 3337, 3503, 4939, 5609, 7093, 7597, 10327, 14987, 20427, 21103, 22345, 22481, 24739, 26491, 27193, 28077, 37753, 37915, 42711, 42717, 47647, 48043, 49243, 50071, 51727, 54823, 57478

Offset: 1

Labos Elemer, Mar 01 2004

nonn

			(sigma(phi(x))-phi(sigma(x)))/sigma(x) quotient equals 1 for x=7, 2 for x=327, 3 for x=5609.

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

is(n)=my(s=sigma(n),t=sigma(eulerphi(n))-eulerphi(s)); t && t%s==0 \\ Charles R Greathouse IV, Feb 14 2013

A065394 Increasing values of A065395: a(n) = A065395(A065393(n)).

Original entry on oeis.org

1, 5, 8, 14, 22, 25, 31, 48, 56, 73, 78, 80, 138, 159, 163, 210, 240, 248, 312, 314, 474, 482, 634, 648, 684, 723, 836, 896, 978, 1026, 1134, 1184, 1320, 1344, 1410, 1424, 1608, 1686, 1760, 1776, 1862, 2226, 2624, 2824, 2936, 3024, 3120, 3280, 3460, 3660

Offset: 1

Labos Elemer, Nov 05 2001

nonn

Amiram Eldar, Table of n, a(n) for n = 1..2000 (terms 1..500 from Harry J. Smith)

Cf. A000010, A000203, A065393, A065395.

a=0; s=0; Do[s=DivisorSigma[1, EulerPhi[n]]-EulerPhi[DivisorSigma[1, n]]; If[s>a, a=s; Print[s]], {n, 1, 10000}]; (* Output is s. *)

{ n=r=0; for (m=1, 10^9, x=sigma(eulerphi(m)) - eulerphi(sigma(m)); if (x > r, r=x; write("b065394.txt", n++, " ", x); if (n==500, return)) ) } \\ Harry J. Smith, Oct 18 2009

a(n) = A000203(A000010(A065393(n))) - A000010(A000203(A065393(n))).

A092586 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

7, 87, 231, 463, 617, 691, 751, 855, 1059, 1127, 2795, 4819, 11999, 18527, 22481, 75311, 121939, 232901, 256751, 288883, 313919, 371519, 845831, 1285841, 1762799, 1815167, 7195199, 9096191, 40324121, 93070943, 99388823, 113140151, 238072223, 487394063

Offset: 1

Labos Elemer, Mar 01 2004

nonn

			(sigma(phi(x))-phi(sigma(x)))/(x+1) equals 1 if x=7; is 2 if x=463; is 3 if x=4819.

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

Edited and extended by Robert G. Wilson v, Mar 03 2004

a(33)-a(34) from Donovan Johnson, Mar 04 2013

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.

Original entry on oeis.org

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

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

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

A092590 a(n) = A065395(A000040(n)); values of commutator of sigma and phi function at prime number arguments.

Original entry on oeis.org

-1, 1, 5, 8, 14, 22, 25, 31, 28, 48, 56, 73, 78, 76, 56, 80, 74, 138, 112, 120, 159, 136, 102, 156, 210, 185, 168, 126, 240, 212, 248, 212, 226, 240, 226, 300, 314, 283, 204, 252, 222, 474, 296, 412, 339, 388, 472, 360, 270, 472, 378, 368, 634, 396, 427, 316, 404, 592, 534, 628, 436, 434, 582, 480, 684, 456, 700, 836

Offset: 1

Labos Elemer, Mar 03 2004

The sequence differs from A065394 since it is not monotonic.

			a(1) = sigma(phi(2))- phi(sigma(2)) = sigma(1)-phi(3) = 1-2 = -1.

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

Cf. A000010, A000040, A000203, A033632, A065393, A065394, A065395, A008331, A008332, A092584, A092585, A092586, A092587, A092588, A092589.

[DivisorSigma(1,EulerPhi(p))-EulerPhi(DivisorSigma(1,p)): p in PrimesUpTo(400)]; // Bruno Berselli, Oct 20 2015

Table[DivisorSigma[1, p-1] - EulerPhi[p+1], {p, Prime[Range[100]]}] (* Amiram Eldar, Jun 09 2024 *)

a(n) = sigma(prime(n)-1) - phi(prime(n)+1) = A008332(n) - A008331(n). - Amiram Eldar, Jun 09 2024

A092591 Exponents m such that 1-A065395(2^m) is a power of 2, where A065395(n) = sigma(phi(n)) - phi(sigma(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 12, 15, 16, 18, 30, 31, 60, 88, 106, 126, 520, 606, 1278, 2202, 2280, 3216, 4252, 4422, 9688, 9940, 11212, 19936, 21700, 23208, 44496, 86242, 110502, 132048, 216090, 756838, 859432, 1257786, 1398268, 2976220, 3021376, 6972592, 13466916, 20996010, 24036582, 25964950, 30402456, 32582656, 37156666, 42643800, 43112608, 57885160

Offset: 1

Labos Elemer, Mar 03 2004

nonn

A000043(k) - 1 is a term for all k >= 1. - Amiram Eldar, Aug 22 2019

Let 2^k = 1-A065395(2^m) = phi(2^(m+1)-1) - 2^m + 2. If k = 0, then phi(2^(m+1)-1) is odd, implying extraneous m = 0. If k = 1, then phi(2^(m+1)-1) = 2^m, meaning that 2^(m+1)-1 is a product of distinct Fermat primes (A019434), which also a term of A050474. The five known Fermat primes give m in {0, 1, 3, 7, 15, 31}. If k >= 2, then phi(2^(m+1)-1) == 2 (mod 4), implying that 2^(m+1)-1 is a prime power, and by Mihăilescu's theorem, 2^(m+1)-1 must be just a prime, that is, m+1 is a term of A000043 and k = m. Hence, unless there exist other Fermat primes, this sequence is the union of {0, 1, 3, 7, 15, 31} and terms of A000043 decreased by 1. - Max Alekseyev, Jun 14 2025

			At exponents m=1, 3, 7, 15, 31: 1-A065395(2^m)=2.
While at m=2, 4, 6, 12, 16, 18, 30, 60, 88, 106, 126: 1-A065395(2^m)=2^m.

Cf. A000010, A000203, A000043, A065395, A092584, A092585, A092586, A092587, A092588, A092589, A092590.

f[n_] := DivisorSigma[1, EulerPhi[n]] - EulerPhi[DivisorSigma[1, n]]; pow2Q[n_] := n == 2^IntegerExponent[n, 2]; aQ[n_] := pow2Q[1 - f[2^n]]; Select[Range[0, 130], aQ] (* Amiram Eldar, Aug 22 2019 *)

f(n) = sigma(eulerphi(n)) - eulerphi(sigma(n)); \\ A065395
ispp2(k) = k == 2^valuation(k,2);
isok(n) = ispp2(1-f(2^n)); \\ Michel Marcus, Aug 22 2019, Jun 16 2025

If there are only 5 Fermat primes (A019434), then for n >= 14, a(n) = A000043(n-5) - 1. - Max Alekseyev, Jun 14 2025

Name and example edited by Michel Marcus, Aug 22 2019
a(18)-a(19) from Amiram Eldar, Aug 23 2019
a(1)=0 inserted and terms a(20) onward added by Max Alekseyev, Jun 14 2025

A092584 Numbers k such that sigma(phi(k)) == phi(sigma(k)) (mod k), that is, A033632(k)/k is an integer.

Original entry on oeis.org

1, 5, 9, 157, 225, 242, 516, 729, 3872, 13932, 14406, 17672, 18225, 20124, 21780, 29262, 29616, 45996, 65025, 76832, 92778, 95916, 106092, 106308, 114630, 114930, 121872, 125652, 140130, 140625, 145794, 149124, 160986, 179562, 185100, 234876

Offset: 1

Labos Elemer, Mar 01 2004

nonn

			Includes but is not identical with A033632.
Below 10^7 only a(2) = 5 and a(4) = 157 give A033632(n)/n nonzero.

Amiram Eldar, Table of n, a(n) for n = 1..2000

Cf. A033632, A092585, A092586, A092587, A092588, A065395.

Select[Range[250000], Divisible[DivisorSigma[1, EulerPhi[#]] - EulerPhi[DivisorSigma[1, #]] , #] &]  (* Amiram Eldar, Mar 12 2020 *)

is(n)=sigma(eulerphi(n))==Mod(eulerphi(sigma(n)),n) \\ Charles R Greathouse IV, Nov 27 2013

A065393 Sigma(phi(m)) - phi(sigma(m)) is increasing at these values of m.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 29, 31, 37, 41, 53, 61, 73, 95, 97, 109, 127, 143, 157, 181, 209, 241, 287, 313, 323, 337, 377, 403, 407, 421, 473, 527, 533, 541, 589, 601, 661, 713, 731, 757, 779, 899, 1009, 1073, 1147, 1159, 1199, 1271, 1321, 1333, 1349, 1517

Offset: 1

Labos Elemer, Nov 05 2001

nonn

First composite number is the 15th term, 95. [Corrected by Jacob Vecht, Jul 28 2020]

Amiram Eldar, Table of n, a(n) for n = 1..2000 (terms 1..500 from Harry J. Smith)

Cf. A000010, A000203, A065394, A065395.

a = 0; s = 0; Do[s = DivisorSigma[1, EulerPhi[n]] - EulerPhi[DivisorSigma[1, n]]; If[s>a, a = s; Print[n]], {n, 1, 10000}]
DeleteDuplicates[Table[{m,DivisorSigma[1,EulerPhi[m]]-EulerPhi[DivisorSigma[1,m]]},{m,1600}],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Aug 02 2023 *)

{ n=r=0; for (m=1, 10^9, x=sigma(eulerphi(m)) - eulerphi(sigma(m)); if (x > r, r=x; write("b065393.txt", n++, " ", m); if (n==500, return)) ) } \\ Harry J. Smith, Oct 18 2009

cp's OEIS Frontend

A092588 Numbers k such that sigma(phi(k)) - phi(sigma(k)) is nonzero and divisible by sigma(k), that is A065395(k)/A000203(k) is a nonzero integer.

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A065394 Increasing values of A065395: a(n) = A065395(A065393(n)).

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A092586 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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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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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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A092589 a(n) = -A065395(2^n).

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A092590 a(n) = A065395(A000040(n)); values of commutator of sigma and phi function at prime number arguments.

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A092591 Exponents m such that 1-A065395(2^m) is a power of 2, where A065395(n) = sigma(phi(n)) - phi(sigma(n)).

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