A066498 -id:A066498 - cp's OEIS frontend

A087943 Numbers n such that 3 divides sigma(n).

2, 5, 6, 8, 10, 11, 14, 15, 17, 18, 20, 22, 23, 24, 26, 29, 30, 32, 33, 34, 35, 38, 40, 41, 42, 44, 45, 46, 47, 49, 50, 51, 53, 54, 55, 56, 58, 59, 60, 62, 65, 66, 68, 69, 70, 71, 72, 74, 77, 78, 80, 82, 83, 85, 86, 87, 88, 89, 90, 92, 94, 95, 96, 98, 99, 101, 102, 104, 105, 106

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 27 2003

nonn

Numbers n such that in the prime factorization n = Product_i p_i^e_i, there is some p_i == 1 (mod 3) with e_i == 2 (mod 3) or some p_i == 2 (mod 3) with e_i odd. - Robert Israel, Nov 09 2016

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000

Cf. A000203, A059269, A066498, A034020, A028983, A074216, A329963 (complement).

select(n -> numtheory:-sigma(n) mod 3 = 0, [$1..1000]); # Robert Israel, Nov 09 2016

Select[Range[1000],Mod[DivisorSigma[1,#],3]==0&] (* Enrique Pérez Herrero, Sep 03 2013 *)

is(n)=sigma(n)%3==0 \\ Charles R Greathouse IV, Sep 04 2013

is(n)=forprime(p=2,997,my(e=valuation(n,p)); if(e && Mod(p,3*p-3)^(e+1)==1, return(1), n/=p^e)); sigma(n)%3==0 \\ Charles R Greathouse IV, Sep 04 2013

a(n) << n^k for any k > 1, where << is the Vinogradov symbol. - Charles R Greathouse IV, Sep 04 2013
a(n) ~ n as n -> infinity: since Sum_{primes p == 2 (mod 3)} 1/p diverges, asymptotically almost every number is divisible by some prime p == 2 (mod 3) but not by p^2. - Robert Israel, Nov 09 2016
Because sigma(n) and sigma(3n)=A144613(n) differ by a multiple of 3, these are also the numbers n such that n divides sigma(3n). - R. J. Mathar, May 19 2020

More terms from Benoit Cloitre and Ray Chandler, Oct 27 2003

A066502 Numbers k such that 7 divides phi(k).

Original entry on oeis.org

29, 43, 49, 58, 71, 86, 87, 98, 113, 116, 127, 129, 142, 145, 147, 172, 174, 196, 197, 203, 211, 213, 215, 226, 232, 239, 245, 254, 258, 261, 281, 284, 290, 294, 301, 319, 337, 339, 343, 344, 348, 355, 377, 379, 381, 387, 392, 394, 406, 421, 422, 426, 430

Offset: 1

Benoit Cloitre, Jan 04 2002

nonn

Related to the equation x^7 == 1 (mod k): sequence gives values of k such there are solutions 1 < x < k of x^7 == 1 (mod k).
If k is a term of this sequence, then G =  is a non-abelian group of order 7k, where 1 < r < n and r^7 == 1 (mod k). For example, G can be the subgroup of GL(2, Z_k) generated by x = {{1, 1}, {0, 1}} and y = {{r, 0}, {0, 1}}. - Jianing Song, Sep 17 2019
The asymptotic density of this sequence is 1 (Dressler, 1975). - Amiram Eldar, May 23 2022

			x^7 == 1 (mod k) has solutions 1 < x < k for k = 29, 43, 49, ...

Harry J. Smith, Table of n, a(n) for n = 1..1000
Robert E. Dressler, A property of the phi and sigma_j functions, Compositio Mathematica, Vol. 31, No. 2 (1975), pp. 115-118.

Cf. A045465, A066498, A066499, A066500, A066501, A000010.

Column k=4 of A277915.

Select[Range[500],Divisible[EulerPhi[#],7]&] (* Harvey P. Dale, Apr 12 2012 *)

isok(k) = { eulerphi(k)%7 == 0 } \\ Harry J. Smith, Feb 18 2010

a(n) are the numbers generated by 7^2 = 49 and all primes congruent to 1 mod 7 (A045465). Hence sequence gives all k such that k == 0 (mod A045465(n)) for some n > 1 or k == 0 (mod 49).

Simpler definition from Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 25 2003

A066500 Numbers k such that 5 divides phi(k).

Original entry on oeis.org

11, 22, 25, 31, 33, 41, 44, 50, 55, 61, 62, 66, 71, 75, 77, 82, 88, 93, 99, 100, 101, 110, 121, 122, 123, 124, 125, 131, 132, 142, 143, 150, 151, 154, 155, 164, 165, 175, 176, 181, 183, 186, 187, 191, 198, 200, 202, 205, 209, 211, 213, 217, 220, 225, 231, 241

Offset: 1

Benoit Cloitre, Jan 04 2002

nonn

Related to the equation x^5 == 1 (mod k): sequence gives values of k such there are solutions 1 < x < k of x^5 == 1 (mod k).
If k is a term of this sequence, then G =  is a non-abelian group of order 5k, where 1 < r < n and r^5 == 1 (mod k). For example, G can be the subgroup of GL(2, Z_k) generated by x = {{1, 1}, {0, 1}} and y = {{r, 0}, {0, 1}}. - Jianing Song, Sep 17 2019
The asymptotic density of this sequence is 1 (Dressler, 1975). - Amiram Eldar, May 23 2022

			x^5 == 1 (mod 11) has solutions 1 < x < 11, namely {3,4,5,9}.

Harry J. Smith, Table of n, a(n) for n = 1..1000
Robert E. Dressler, A property of the phi and sigma_j functions, Compositio Mathematica, Vol. 31, No. 2 (1975), pp. 115-118.

Cf. A000010, A045453, A066498, A066499, A066501, A066502.

Column k=3 of A277915.

Select[Range[250], Divisible[EulerPhi[#], 5] &] (* Amiram Eldar, May 23 2022 *)

isok(k) = { eulerphi(k)%5 == 0 } \\ Harry J. Smith, Feb 18 2010

a(n) are the numbers generated by 5^2 = 25 and all primes congruent to 1 mod 5 (A045453). Hence sequence gives all k such that k == 0 (mod A045453(n)) for some n > 1 or k == 0 (mod 25).

Simpler definition from Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 25 2003

Extended by Ray Chandler, Nov 06 2003

A088232 Numbers k such that 3 does not divide phi(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 15, 16, 17, 20, 22, 23, 24, 25, 29, 30, 32, 33, 34, 40, 41, 44, 46, 47, 48, 50, 51, 53, 55, 58, 59, 60, 64, 66, 68, 69, 71, 75, 80, 82, 83, 85, 87, 88, 89, 92, 94, 96, 100, 101, 102, 106, 107, 110, 113, 115, 116, 118, 120, 121, 123, 125, 128

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 03 2003

nonn

n such that the congruence x^3 == 1 mod(n) has only the trivial solution x=1 i.e. A060839(n) = 1 . Complement of sequence A066498.
Let U(n) be the group of positive integers coprime to n under mod n multiplication.  Let U(n)^3 = {x^3: x is an element of U(n)}.  These are the n such that U(n) = U(n)^3. - Geoffrey Critzer, Jun 07 2015
In other words, numbers divisible neither by 9 nor by any primes of the form 6k+1. - Ivan Neretin, Sep 03 2015
The asymptotic density of this sequence is 0 (Dressler, 1975). - Amiram Eldar, Jul 23 2020

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Robert E. Dressler, A property of the phi and sigma_j functions, Compositio Mathematica, Vol. 31, No. 2 (1975), pp. 115-118.
Kevin Ford, Florian Luca, and Pieter Moree, Values of the Euler phi-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields, arXiv:1108.3805 [math.NT], 2011-2012.

Cf. A000010, A066498 (complement).
Positions of 1's in A060839, of 0's in A354099, of nonzeros in A074942.
Cf. also A329963.

select(t -> numtheory:-phi(t) mod 3 <> 0, [$1..1000]); # Robert Israel, Sep 04 2015

Prepend[Position[Table[Union[Select[Range[n], CoprimeQ[#, n] &]] ==
     Union[Mod[Select[Range[n], CoprimeQ[#, n] &]^3, n]], {n, 1,155}], True], 1] // Flatten (* Geoffrey Critzer, Jun 07 2015 *)
Select[Range[140],!Divisible[EulerPhi[#],3]&] (* Harvey P. Dale, Sep 23 2017 *)

is(n)=eulerphi(n)%3 \\ Charles R Greathouse IV, Feb 04 2013

a(n) ~ k n sqrt(log(n)) for some constant k. k appears to be around 1.08. [Charles R Greathouse IV, Feb 14 2012]

More terms from Ray Chandler, Nov 05 2003

A066499 Numbers k such that phi(k) == 2 (mod 4).

Original entry on oeis.org

3, 4, 6, 7, 9, 11, 14, 18, 19, 22, 23, 27, 31, 38, 43, 46, 47, 49, 54, 59, 62, 67, 71, 79, 81, 83, 86, 94, 98, 103, 107, 118, 121, 127, 131, 134, 139, 142, 151, 158, 162, 163, 166, 167, 179, 191, 199, 206, 211, 214, 223, 227, 239, 242, 243, 251, 254, 262, 263, 271

Offset: 1

Benoit Cloitre, Jan 04 2002

nonn

Related to the equation x^4 = 1 (mod y): sequence gives values of n such x^4 = 1 (mod n) has no solution 1 < x < n-1.
k is of the form p^m or 2*p^m where p is A002145 (with the exception of a(2)=4).
All prime numbers here belong also to A002145, prime numbers of the form 4n+3. - Enrique Pérez Herrero, Sep 07 2011

W. J. LeVeque, Fundamentals of Number Theory, pp. 57 Problem 15, Dover NY 1996.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A066498, A066500, A066501, A066502, A000010.
Essentially the same as A097987.
Cf. A002145.

Select[Range[300],Mod[EulerPhi[#],4]==2&] (* Harvey P. Dale, Feb 18 2018 *)

isok(k) = { eulerphi(k)%4 == 2 } \\ Harry J. Smith, Feb 18 2010

Simpler definition from Lekraj Beedassy, Jul 21 2003

Corrected and extended by Ray Chandler, Nov 06 2003

A277915 A(n,k) is the n-th number m such that a nontrivial prime(k)-th root of unity modulo m exists; square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

8, 7, 12, 11, 9, 15, 29, 22, 13, 16, 23, 43, 25, 14, 20, 53, 46, 49, 31, 18, 21, 103, 79, 67, 58, 33, 19, 24, 191, 137, 106, 69, 71, 41, 21, 28, 47, 229, 206, 131, 89, 86, 44, 26, 30, 59, 94, 361, 239, 157, 92, 87, 50, 27, 32, 311, 118, 139, 382, 274, 158, 115, 98, 55, 28, 33

Offset: 1

Alois P. Heinz, Nov 03 2016

The trivial square roots of unity modulo m are {1, m-1} and for an odd prime p the trivial p-th root of unity modulo m is 1.
There is no prime in the first column.
Column k>1 contains prime(k)^2.

			Square array A(n,k) begins:
:  8,  7, 11, 29,  23,  53, 103, 191, ...
: 12,  9, 22, 43,  46,  79, 137, 229, ...
: 15, 13, 25, 49,  67, 106, 206, 361, ...
: 16, 14, 31, 58,  69, 131, 239, 382, ...
: 20, 18, 33, 71,  89, 157, 274, 419, ...
: 21, 19, 41, 86,  92, 158, 289, 457, ...
: 24, 21, 44, 87, 115, 159, 307, 458, ...
: 28, 26, 50, 98, 121, 169, 309, 571, ...

Alois P. Heinz, Antidiagonals n = 1..200, flattened
Wikipedia, Root of unity modulo n

Columns k=1-4 give: A033949, A066498, A066500, A066502.
Row n=1 gives A066674 for k>1.
Main diagonal gives A305828.
Cf. A000010, A000040, A002322.

with(numtheory):
A:= proc() local j, l; l:= proc() [] end;
      proc(n, k)
        while nops(l(k)) lambda(j) or k>1 and
                  irem(phi(j), ithprime(k))=0 then
                  l(k):= [l(k)[], j]; break fi
          od
        od: l(k)[n]
      end
    end():
seq(seq(A(n, 1+d-n), n=1..d), d=1..15);

A[n_, k_] := Module[{j, l = {}}, While[Length[l]CarmichaelLambda[j] || k>1 && Mod[EulerPhi[j], Prime[k]]==0, AppendTo[l, j]; Break[]]]]; l[[n]]];
Table[A[n, 1 + d - n], {d, 1, 15}, {n, 1, d}] // Flatten (* Jean-François Alcover, May 29 2018, from Maple *)

A066501 Numbers k such that x^6 == 1 (mod(k)) has no solution 1 < x < k-1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 10, 11, 17, 22, 23, 25, 29, 34, 41, 46, 47, 50, 53, 58, 59, 71, 82, 83, 89, 94, 101, 106, 107, 113, 118, 121, 125, 131, 137, 142, 149, 166, 167, 173, 178, 179, 191, 197, 202, 214, 226, 227, 233, 239, 242, 250, 251, 257, 262, 263, 269, 274, 281, 289, 293

Offset: 1

Benoit Cloitre, Jan 04 2002

nonn

Cf. A045309, A066498, A066499, A066500, A066502, A000010.

isok(n) = {for (x=2, n-2, if ((Mod(x, n)^6) == Mod(1, n), return (0));); return (1);} \\ Michel Marcus, Nov 20 2013

Sequence consists of the numbers 4, 6 and for all k > 1, A045309(k), 2*A045309(k), A045309(k)^2, 2*A045309(k)^2.

Extended by Ray Chandler, Nov 06 2003

Terms 1, 2 and 3 prepended by Michel Marcus, Nov 20 2013

A172019 Numbers k such that 4 divides phi(k) (i.e., A000010(k)).

Original entry on oeis.org

5, 8, 10, 12, 13, 15, 16, 17, 20, 21, 24, 25, 26, 28, 29, 30, 32, 33, 34, 35, 36, 37, 39, 40, 41, 42, 44, 45, 48, 50, 51, 52, 53, 55, 56, 57, 58, 60, 61, 63, 64, 65, 66, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 82, 84, 85, 87, 88, 89, 90, 91, 92, 93, 95, 96, 97, 99, 100, 101

Offset: 1

Giovanni Teofilatto, Jan 22 2010

Complement of A097987.

The asymptotic density of this sequence is 1 (Dressler, 1975). - Amiram Eldar, Feb 12 2021

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Robert E. Dressler, A property of the phi and sigma_j functions, Compositio Mathematica, Vol. 31, No. 2 (1975), pp. 115-118.

Cf. A000010, A097987, A066499, A066498, A066500, A066502.

Select[Range[200], Mod[EulerPhi[#], 4] == 0 &] (* Geoffrey Critzer, Nov 30 2014 *)

is(n)=my(o=valuation(n, 2), p); (o>1 || !isprimepower(n>>o, &p) || p%4<2) && n>4 \\ Charles R Greathouse IV, Mar 05 2013

A354099 The 3-adic valuation of Euler totient function phi.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 2, 0, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 1, 0, 0, 0, 1, 1, 2, 2, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 1, 2, 0, 0, 0, 1, 1, 2, 0, 1, 0, 1, 0, 0, 1, 0, 1, 2, 2, 0, 2, 1, 1, 1, 0, 3, 0, 0, 1, 0, 1, 0, 0, 0, 1, 2, 0, 1, 0, 2, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1

Offset: 1

Antti Karttunen, May 17 2022

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A007949, A074942.
Cf. A088232 (positions of zeros), A066498 (of terms > 0).
Cf. also A354100.

a[n_] := IntegerExponent[EulerPhi[n], 3]; Array[a, 100] (* Amiram Eldar, May 17 2022 *)

PARI
```
A354099(n) = valuation(eulerphi(n),3);
```

A354099(n) = { my(f=factor(n)); sum(k=1,#f~,valuation((f[k,1]-1)*(f[k,1]^(f[k,2]-1)), 3)); }; \\ Demonstrates the additivity.

a(n) = A007949(A000010(n)).

Additive with a(p^e) = A007949((p-1)*p^(e-1)).

A254073 Number of solutions to x^3 + y^3 + z^3 == 1 (mod n) for 1 <= x, y, z <= n.

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 90, 64, 162, 100, 121, 144, 252, 360, 225, 256, 289, 648, 468, 400, 810, 484, 529, 576, 625, 1008, 1458, 1440, 841, 900, 1143, 1024, 1089, 1156, 2250, 2592, 1602, 1872, 2268, 1600, 1681, 3240, 2115, 1936, 4050, 2116, 2209, 2304, 4410

Offset: 1

José María Grau Ribas, Jan 25 2015

It appears that a(n) = n^2 for n in A088232 (numbers n such that 3 does not divide phi(n)) and that a(n) != n^2 for n in A066498 (numbers n such that 3 divides phi(n)). - Michel Marcus, Mar 13 2015

It appears that a(p) != p^2 for primes in A002476 (primes of form 6m + 1). - Michel Marcus, Mar 13 2015

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A087412.

a[n_] := Sum[ If[ Mod[x^3 + y^3 + z^3, n] == 1, 1, 0], {x, n}, {y, n}, {z, n}]; a[1]=1; Table[a[n], {n, 2,22}]

a(n) = {nb = 0; for (x=1, n, for (y=1, n, for (z=1, n, if ((Mod(x^3,n) + Mod(y^3,n) + Mod(z^3,n)) % n == Mod(1, n), nb ++);););); nb;} \\ Michel Marcus, Mar 11 2015

a(n)={my(p=Mod(sum(i=0, n-1, x^(i^3%n)), 1-x^n)^3); polcoeff(lift(p), 1%n)} \\ Andrew Howroyd, Jul 18 2018

def A254073(n):
    ndict = {}
    for i in range(n):
        m = pow(i,3,n)
        if m in ndict:
            ndict[m] += 1
        else:
            ndict[m] = 1
    count = 0
    for i in ndict:
        ni = ndict[i]
        for j in ndict:
            k = (1-i-j) % n
            if k in ndict:
                count += ni*ndict[j]*ndict[k]
    return count # Chai Wah Wu, Jun 06 2017

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A087943 Numbers n such that 3 divides sigma(n).

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A066502 Numbers k such that 7 divides phi(k).

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A066500 Numbers k such that 5 divides phi(k).

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A088232 Numbers k such that 3 does not divide phi(k).

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A066499 Numbers k such that phi(k) == 2 (mod 4).

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A277915 A(n,k) is the n-th number m such that a nontrivial prime(k)-th root of unity modulo m exists; square array A(n,k), n>=1, k>=1, read by antidiagonals.

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A066501 Numbers k such that x^6 == 1 (mod(k)) has no solution 1 < x < k-1.

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