A083533 -id:A083533 - cp's OEIS frontend

A383889 Record high points in A083533.

1, 2, 4, 6, 10, 12, 16, 18, 20, 22, 24, 26, 28, 32, 36, 40, 44, 50, 60, 64, 72, 74, 76, 78, 80, 90, 96, 108, 112

Offset: 1

Jud McCranie, May 13 2025

Values of A083533(k) such that A083533(k) > A083533(i) for any iA002202(j+1) - A002202(j) > A002202(i+1) - A002202(i) for all iA383890 has the corresponding index of A083533.

Cf. A000010, A002202, A083533, A383890.

A002202 Values taken by totient function phi(m) (A000010).

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 28, 30, 32, 36, 40, 42, 44, 46, 48, 52, 54, 56, 58, 60, 64, 66, 70, 72, 78, 80, 82, 84, 88, 92, 96, 100, 102, 104, 106, 108, 110, 112, 116, 120, 126, 128, 130, 132, 136, 138, 140, 144, 148, 150, 156, 160, 162, 164, 166, 168, 172, 176

Offset: 1

N. J. A. Sloane

These are the numbers n such that for some m the multiplicative group mod m has order n.
Maier & Pomerance show that there are about x * exp(c (log log log x)^2)/log x members of this sequence up to x, with c = 0.81781465... (A234614); see the paper for details on making this precise. - Charles R Greathouse IV, Dec 28 2013
A264739(a(n)) = 1; a(n) occurs A058277(n) times in A007614. - Reinhard Zumkeller, Nov 26 2015
There are no odd numbers > 2 in the sequence and the even numbers that are not in the sequence are in A005277. - Bernard Schott, May 13 2020

J. W. L. Glaisher, Number-Divisor Tables. British Assoc. Math. Tables, Vol. 8, Camb. Univ. Press, 1940, p. 64.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
André Contiero, and Davi Lima, 2-Adic Stratification of Totients, arXiv:2005.05475 [math.NT], 2020.
K. Ford, The distribution of totients, Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 27-34.
Helmut Maier and Carl Pomerance, On the number of distinct values of Euler's phi-function, Acta Arithmetica 49:3 (1988), pp. 263-275.
Maxim Rytin, Finding the Inverse of Euler Totient Function (1999).
S. Sivasankaranarayana Pillai, On some functions connected with phi(n), Bull. Amer. Math. Soc. 35 (1929), 832-836.
Eric Weisstein's World of Mathematics, Totient Valence Function

Cf. A000010, A002180, A032446, A058277.
Cf. A002110, A005277, A007614, A007617 (complement).
Cf. A083533 (first differences), A264739.
Cf. A006093 (a subsequence).

import Data.List.Ordered (insertSet)
a002202 n = a002202_list !! (n-1)
a002202_list = f [1..] (tail a002110_list) [] where
   f (x:xs) ps'@(p:ps) us
     | x < p = f xs ps' $ insertSet (a000010' x) us
     | otherwise = vs ++ f xs ps ws
     where (vs, ws) = span (<= a000010' x) us
-- Reinhard Zumkeller, Nov 22 2015

with(numtheory); t1 := [seq(nops(invphi(n)), n=1..300)]; t2 := []: for n from 1 to 300 do if t1[n] <> 0 then t2 := [op(t2), n]; fi; od: t2;
# second Maple program:
q:= n-> is(numtheory[invphi](n)<>[]):
select(q, [$1..176])[];  # Alois P. Heinz, Nov 13 2024

phiQ[m_] := Select[Range[m+1, 2m*Product[(1-1/(k*Log[k]))^(-1), {k, 2, DivisorSigma[0, m]}]], EulerPhi[#] == m &, 1 ] != {}; Select[Range[176], phiQ] (* Jean-François Alcover, May 23 2011, after Maxim Rytin *)

lst(lim)=my(P=1,q,v);forprime(p=2,default(primelimit), if(eulerphi(P*=p)>=lim,q=p;break));v=vecsort(vector(P/q*lim\eulerphi(P/q),k,eulerphi(k)),,8);select(n->n<=lim,v) \\ Charles R Greathouse IV, Apr 16 2012

select(istotient,vector(100,i,i)) \\ Charles R Greathouse IV, Dec 28 2012

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 *)

A083534 First difference sequence of A007617. Difference between consecutive values not being in the range of phi (A000010).

Original entry on oeis.org

2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2

Offset: 1

Labos Elemer, May 20 2003

nonn

a(n) is either 2 or 1 since odd numbers are in A007619.

If a(n) = 1 then A007619(n+1) is an even number not in the range of phi.

			{11,13,14,15,17} are not in the range of phi and the corresponding differences are {2,1,1,2}.

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

Cf. A000010, A002202, A005277, A007617, A007619, A083531, A083532, A083533, A083535, A083536.

a083534 n = a083534_list !! (n-1)
a083534_list = zipWith (-) (tail a007617_list) a007617_list
-- Reinhard Zumkeller, Nov 26 2015

t0[x_] := Table[j, {j, 1, x}]; t=Table[EulerPhi[w], {w, 1, 10000}]; u=Union[%]; c=Complement[t0[10000], u]; Delete[c-RotateRight[c], 1]

list(lim) = {my(k1 = 3, k2 = 3); while(k1 < lim, until(!istotient(k2), k2++); print1(k2 - k1, ", "); k1 = k2); } \\ Amiram Eldar, Feb 22 2025

a(n) = A007617(n+1) - A007617(n).

A333019 Numbers k such that both k and k + 2 are totient numbers (A002202).

Original entry on oeis.org

2, 4, 6, 8, 10, 16, 18, 20, 22, 28, 30, 40, 42, 44, 46, 52, 54, 56, 58, 64, 70, 78, 80, 82, 100, 102, 104, 106, 108, 110, 126, 128, 130, 136, 138, 148, 160, 162, 164, 166, 176, 178, 190, 196, 198, 208, 210, 220, 222, 224, 226, 238, 250, 260, 262, 268, 270, 280

Offset: 1

Amiram Eldar, Mar 05 2020

nonn

			2 is a term since both 2 and 4 are totient numbers.

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

Cf. A000010, A002202, A005277, A063512, A083533, A333020, A333021, A333022, A333023, A333024.

for(k = 1, 150, if(istotient(2*k) && istotient(2*k+2), print1(2*k,", ")))

A333020 Starts of runs of 3 consecutive even numbers that are all totient numbers (A002202).

Original entry on oeis.org

2, 4, 6, 8, 16, 18, 20, 28, 40, 42, 44, 52, 54, 56, 78, 80, 100, 102, 104, 106, 108, 126, 128, 136, 160, 162, 164, 176, 196, 208, 220, 222, 224, 260, 268, 292, 328, 342, 344, 356, 378, 380, 416, 438, 440, 460, 462, 464, 476, 498, 500, 502, 504, 520, 560, 584

Offset: 1

Amiram Eldar, Mar 05 2020

nonn

			2 is a term since 2, 4 and 6 are all totient numbers.

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

Cf. A000010, A002202, A005277, A063512, A083533, A333019, A333021, A333022, A333023, A333024.

m = 3; v = vector(m); for(k=1, m, v[k] = istotient(2*k)); for(k = m+1, 300, if(Set(v) == [1], print1(2*(k-m),", ")); v = concat(v[2..m], istotient(2*k)))

A333021 Starts of runs of 4 consecutive even numbers that are all totient numbers (A002202).

Original entry on oeis.org

2, 4, 6, 16, 18, 40, 42, 52, 54, 78, 100, 102, 104, 106, 126, 160, 162, 220, 222, 342, 378, 438, 460, 462, 498, 500, 502, 856, 858, 880, 882, 1086, 1276, 1278, 1300, 1422, 1480, 1482, 1566, 1660, 1662, 1804, 1806, 1996, 2058, 2200, 2202, 2236, 2238, 3016, 3018

Offset: 1

Amiram Eldar, Mar 05 2020

nonn

			2 is a term since 2, 4, 6 and 8 are all totient numbers.

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

Cf. A000010, A002202, A005277, A063512, A083533, A333019, A333020, A333022, A333023, A333024.

m = 4; v = vector(m); for(k=1, m, v[k] = istotient(2*k)); for(k = m+1, 1500, if(Set(v) == [1], print1(2*(k-m),", ")); v = concat(v[2..m], istotient(2*k)))

A333022 Starts of runs of 5 consecutive even numbers that are all totient numbers (A002202).

Original entry on oeis.org

2, 4, 16, 40, 52, 100, 102, 104, 160, 220, 460, 498, 500, 856, 880, 1276, 1480, 1660, 1804, 2200, 2236, 3016, 3160, 3460, 4516, 4780, 5500, 5920, 6040, 6196, 6820, 7240, 7636, 7696, 7720, 8536, 8620, 9196, 9460, 9880, 10456, 12916, 13756, 13960, 14416, 15640

Offset: 1

Amiram Eldar, Mar 05 2020

nonn

			2 is a term since 2, 4, 6, 8 and 10 are all totient numbers.

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

Cf. A000010, A002202, A005277, A063512, A083533, A333019, A333020, A333021, A333023, A333024.

m = 5; v = vector(m); for(k=1, m, v[k] = istotient(2*k)); for(k = m+1, 7500, if(Set(v) == [1], print1(2*(k-m),", ")); v = concat(v[2..m], istotient(2*k)))

A333023 Starts of runs of 6 consecutive even numbers that are all totient numbers (A002202).

Original entry on oeis.org

2, 100, 102, 498, 267670, 26734060, 26734062, 31253680, 65974998, 70938496, 118428800, 1232747200, 2764919296, 3149734998, 3149735000, 3413655896, 3415058276, 3755544796, 4446555802, 5727840798, 6156991616, 10080661998, 10464983096, 11054945296, 11953158220

Offset: 1

Amiram Eldar, Mar 05 2020

nonn

			2 is a term since 2, 4, 6, 8, 10 and 12 are all totient numbers.

Amiram Eldar, Table of n, a(n) for n = 1..46
Wikipedia, Totient numbers.

Cf. A000010, A002202, A005277, A063512, A083533, A333019, A333020, A333021, A333022, A333024.

m = 6; v = vector(m); for(k=1, m, v[k] = istotient(2*k)); for(k = m+1, 1.5e5, if(Set(v) == [1], print1(2*(k-m),", ")); v = concat(v[2..m], istotient(2*k)))

cp's OEIS Frontend

A383889 Record high points in A083533.

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A002202 Values taken by totient function phi(m) (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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A083534 First difference sequence of A007617. Difference between consecutive values not being in the range of phi (A000010).

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A333019 Numbers k such that both k and k + 2 are totient numbers (A002202).

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A333020 Starts of runs of 3 consecutive even numbers that are all totient numbers (A002202).

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A333021 Starts of runs of 4 consecutive even numbers that are all totient numbers (A002202).

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