A092585 -id:A092585

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A092584

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

+10
8

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

EXAMPLE

Includes but is not identical with A033632.

Below 10^7 only a(2) = 5 and a(4) = 157 give A033632(n)/n nonzero.

MATHEMATICA

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

PROG

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Labos Elemer, Mar 01 2004

STATUS

approved

A092589

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

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

FORMULA

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

MATHEMATICA

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

CROSSREFS

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

KEYWORD

sign

AUTHOR

Labos Elemer, Mar 02 2004

EXTENSIONS

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

STATUS

approved

A092590

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

+10
2

-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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

The sequence differs from A065394 since it is not monotonic.

LINKS

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

FORMULA

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

EXAMPLE

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

MATHEMATICA

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

PROG

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

CROSSREFS

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

KEYWORD

sign,look

AUTHOR

Labos Elemer, Mar 03 2004

STATUS

approved

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