A092586 -id:A092586 - cp's OEIS frontend

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

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

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

A341534 Number of possible final configurations in a biased cake-cutting procedure for n people.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 7, 8, 11, 12, 13, 14, 16, 18, 19, 20, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 35, 37, 38, 39, 40, 41, 43, 45, 47, 48, 49, 50, 51, 52, 53, 55, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 71, 72, 74, 75, 76, 77, 78, 79, 80, 81

Offset: 1

Luc Rousseau, Feb 13 2021

nonn

A cake of size 1 is to be divided between n people. The cutter iterates the following procedure: to obtain a new piece of the cake, he cuts the biggest piece into two subpieces (if several pieces are biggest ex aequo, he cuts just one of them); the cut is biased in the proportions t versus 1-t, where t is a constant real number between 1/2 and 1. We will assume that t is transcendental. After the procedure is applied, each piece's size is clearly t^x*(1-t)^y for some (x,y), ordered pair of nonnegative integers. The obtained ordered multiset of t^x*(1-t)^y polynomials depends on t and n; we shall call this the f(t,n) configuration. By definition a(n) is Card({f(t,n); t varies}), i.e., the number of configurations that the procedure for n people can possibly generate, when one does not know the value of t before it starts.

The problem boils down to a geometrical one, where one has to compare the Y-coordinates of rotated lattice points, the angle of rotation depending on t: tan(angle)=log(1-t)/log(t). See SVG link.

			=========================================================================
  1 |    2    |          3          |                4                  |
=========================================================================
    |         |                     | t^3 > 1-t > t*(1-t) > t^2*(1-t)   |
    |         | t^2 > 1-t > t*(1-t) +-----------------------------------+
    |         |                     | 1-t > t^3 > t*(1-t) > t^2*(1-t)   |
  1 | t > 1-t +---------------------+-----------------------------------+
    |         | 1-t > t^2 > t*(1-t) | t^2 > t*(1-t) = t*(1-t) > (1-t)^2 |
=========================================================================
n=1: the whole cake is the only piece, a(1) = 1.
n=2: the first division necessarily divided 1 into t and 1-t; t and 1-t are necessarily ordered this way: t > 1-t. a(2) = 1.
n=3: the second division necessarily divided t into t^2 and t*(1-t); t*(1-t) is necessarily smaller than both t^2 and 1-t; but either t^2 or 1-t may be the biggest: it depends on whether t < 1/phi or t > 1/phi, where phi denotes the golden ratio; so there are two cases and a(3) = 2.
n=4:
  * in the case when t^2 > 1-t, the third division divided t^2 into t^3 and t^2*(1-t); t^2*(1-t) is necessarily smaller than t*(1-t) which is necessarily smaller than both t^3 and 1-t; but either t^3 or 1-t may be the biggest: it depends on whether t < 1/psi or t > 1/psi, where psi denotes the constant described in A092586 (sometimes called the supergolden ratio); so there are two subcases;
  * in the case when 1-t > t^2, the third division divided 1-t into t*(1-t) and (1-t)^2; the order of the elements is fully determined without requiring new assumptions on t, so there is just one subcase;
  * gathering all subcases contributions yields a(4) = 3.

Luc Rousseau, Java program
Luc Rousseau, SVG illustration for n=1..10

Cf. A094214, A001622 (1/phi, phi).

Cf. A263719, A092586 (1/psi, psi).

Java
```
// See Rousseau link.
```

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

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A341534 Number of possible final configurations in a biased cake-cutting procedure for n people.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Java

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions