A083534 -id:A083534 - cp's OEIS frontend

A007617 Values not in range of Euler phi function.

3, 5, 7, 9, 11, 13, 14, 15, 17, 19, 21, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 45, 47, 49, 50, 51, 53, 55, 57, 59, 61, 62, 63, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 81, 83, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97, 98, 99, 101, 103, 105, 107

Offset: 1

Walter Nissen

nonn

Nontotient numbers.
All odd numbers > 2 are in the sequence.
The even numbers of the sequence are in A005277.
The asymptotic density of this sequence is 1. - Amiram Eldar, Mar 26 2021

			There are no solutions to phi(m)=14, so 14 is a member of the sequence.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, section B36, page 138-142.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Jerzy Browkin and Andrzej Schinzel, On integers not of the form n-phi(n), Colloq. Math., Vol. 58 (1995), pp. 55-58.
Paul Erdős and R. R. Hall, Distinct values of Euler's phi-function, Mathematika, Vol. 23 (1976), pp. 1-3.
Kevin Ford, The distribution of totients. Paul Erdős (1913-1996). Ramanujan J., Vol. 2 (1998) pp. 67-151; arXiv preprint, arXiv:1104.3264 [math.NT], 2011-2013.
Kevin Ford, The distribution of totients, Electron. Res. Announc. Amer. Math. Soc., Vol. 4 (1998) pp. 27-34.
Kevin Ford, The number of solutions of phi(x)=m, Ann. of Math.(2), Vol. 150, No. 1 (1999), pp. 283-311.
Helmut Maier and Carl Pomerance, On the number of distinct values of Euler's phi-function, Acta Arithmetica, Vol. 49, No. 3 (1988), pp. 263-275.
Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
Maxim Rytin, Finding the Inverse of Euler Totient Function, Wolfram Library Archive, 1999.
Zhang Ming-Zhi, On nontotients, J. Number Theory, Vol. 43, No. 2 (1993), pp. 168-173.

Numbers not in A000010.
Complement of A002202.
Cf. A005277, A180639.
Cf. A083534 (first differences), A264739.

import Data.List.Ordered (minus)
a007617 n = a007617_list !! (n-1)
a007617_list = [1..] `minus` a002202_list
-- Reinhard Zumkeller, Nov 22 2015

A007617 := n -> if invphi(n)=[] then n fi: seq(A007617(i),i=1..107); # Peter Luschny, Jun 26 2011

inversePhi[m_?OddQ] = {}; inversePhi[1] = {1, 2}; inversePhi[m_] := Module[{p, nmax, n, nn}, p = Select[Divisors[m] + 1, PrimeQ]; nmax = m*Times @@ (p/(p - 1)); n = m; nn = {}; While[n <= nmax, If[EulerPhi[n] == m, AppendTo[nn, n]]; n++]; nn]; Select[Range[107], inversePhi[#] == {} &] (* Jean-François Alcover, Jan 03 2012 *)
Select[Range[107], invphi[#] == {}&] (* Jean-François Alcover, Mar 19 2019, using Maxim Rytin's much faster 'invphi' program *)

is(n)=!istotient(n) \\ Charles R Greathouse IV, Dec 28 2013

A264739(a(n)) = 0. - Reinhard Zumkeller, Nov 26 2015

A083533 First difference sequence of A002202. Difference between consecutive possible values of phi(n), the Euler totient function A000010.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 2, 2, 4, 4, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 2, 4, 2, 6, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 4, 4, 6, 2, 2, 2, 4, 2, 2, 4, 4, 2, 6, 4, 2, 2, 2, 2, 4, 4, 2, 2, 4, 6, 2, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 2, 2, 4, 6, 2, 10, 2, 4, 4, 2, 2, 4, 2, 2, 4, 4, 2, 6, 4, 2, 2, 4, 6, 4, 2, 4

Offset: 1

Labos Elemer, May 20 2003

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000010, A002202, A005277, A083531, A083532, A083534, A083535, A083536.

a083533 n = a083533_list !! (n-1)
a083533_list = zipWith (-) (tail a002202_list) a002202_list
-- Reinhard Zumkeller, Nov 26 2015

t=Table[EulerPhi[w], {w, 1, 25000}]; u=Union[%]; Delete[u-RotateRight[u], 1]

lista(lim) = {my(k1 = 1, k2 = 1); while(k1 < lim, until(istotient(k2), k2++); print1(k2 - k1, ", "); k1 = k2);} \\ Amiram Eldar, Nov 16 2024

a(n) = A002202(n+1) - A002202(n).

A063742 Cototients: numbers k such that x - phi(x) = k has at least one solution.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78

Offset: 1

Labos Elemer, Aug 13 2001, corrected May 20 2003

nonn

Numbers in range of A051953, the cototients. Complement of non-cototients, A005278 with respect to natural numbers A000027.

			First missing numbers are: 10,26,34,50,52,... These missing values are collected in A005278.

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

Cf. A000010, A005278, A051953, A083533, A083534, A083536.

Analogous to A002202.

Union[Table[w-EulerPhi[w], {w, 1, A}]] (* taken for sufficiently large A. *)

Edited by N. J. A. Sloane, Nov 17 2008 at the suggestion of R. J. Mathar

A083531 First difference sequence of A002191. Differences between possible values for sum of divisors of n.

Original entry on oeis.org

2, 1, 2, 1, 1, 4, 1, 1, 1, 3, 2, 4, 4, 2, 1, 1, 4, 2, 1, 1, 2, 2, 4, 6, 2, 1, 3, 2, 1, 5, 4, 2, 4, 2, 4, 6, 1, 2, 3, 2, 4, 2, 4, 2, 2, 2, 6, 1, 3, 2, 1, 1, 4, 1, 5, 2, 4, 6, 2, 4, 2, 2, 2, 2, 4, 3, 3, 2, 4, 2, 1, 3, 6, 2, 1, 3, 2, 4, 6, 2, 4, 1, 5, 2, 4, 2, 4, 6, 2, 6, 4, 3, 1, 2, 2, 4, 2, 4, 2, 6, 2, 2, 2, 4, 6

Offset: 1

Labos Elemer, May 20 2003

nonn

			8 and 12 are the 6th and 7th possible values for sigma(x), since they are sum of divisors of x = 7 and x = 11 respectively, while 9, 10, 11 are impossible ones so 12 - 8 = 4 = a(6) = A002191(7) - A002191(6).
From _Michael De Vlieger_, Jul 22 2017: (Start)
First position of values:
Value   First position
    1         2
    2         1
    3        10
    4         6
    5        30
    6        24
    7       277
    8       165
    9       509
   10       150
   11       824
   12       400
   13     10970
   14      1400
   15     10448
   16      1182
   17     18731
   18      2218
   19    209237
   20      3420
   21    127385
   22      6910
   23     28899
   24      5377
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A002191, A007609, A007369, A083532, A083533, A083534, A083535, A083536, A109323 (start of record gaps in A002191).

t=Table[DivisorSigma[1, w], {w, 1, 25000}]; u=Union[%]; Delete[u-RotateRight[u], 1]
(* Second program: *)
With[{nn = 300}, Differences@ TakeWhile[Union@ DivisorSigma[1, Range@ nn], # < nn &]] (* Michael De Vlieger, Jul 22 2017 *)

A083532 First difference sequence of A007369. Differences between impossible values for sum of divisors of n.

Original entry on oeis.org

3, 4, 1, 1, 5, 1, 2, 2, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 4, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 3, 1, 2, 4, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 1, 2, 1

Offset: 1

Labos Elemer, May 20 2003

nonn

			29 and 33 are the 15th and 16th nonsense values for sigma(x), since there exist no numbers n of which they are sums of divisors, while {30,31,32} equal sigma(x); e.g., for x = 29, 16, 31, respectively, thus 33 - 29 = 4 = a(15) = A007369(16) - A007369(15).

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

Cf. A002191, A007369, A083235, A083531, A083533, A083534, A083536.

t0[x_] := Table[j, {j, 1, x}]; t=Table[DivisorSigma[1, w], {w, 1, 25000}]; u=Union[%]; c=Complement[t0[25000], u]; Delete[c-RotateRight[c], 1]

a(n) = A007369(n+1) - A007369(n).

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A007617 Values not in range of Euler phi function.

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A083533 First difference sequence of A002202. Difference between consecutive possible values of phi(n), the Euler totient function A000010.

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A063742 Cototients: numbers k such that x - phi(x) = k has at least one solution.

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A083531 First difference sequence of A002191. Differences between possible values for sum of divisors of n.

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A083532 First difference sequence of A007369. Differences between impossible values for sum of divisors of n.

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