A078330 -id:A078330 - cp's OEIS frontend

A089451 a(n) = mu(prime(n)-1), where mu is the Moebius function (A008683).

1, -1, 0, 1, 1, 0, 0, 0, 1, 0, -1, 0, 0, -1, 1, 0, 1, 0, -1, -1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 1, 0, 1, 0, -1, 0, 0, 0, 1, -1, 1, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 1, 0, 1, 0, 0, 1, -1, 0, 0, 1, 0, 0, 0, 0, -1, 0, -1, 0, -1, -1, 0, 0, 0, 1, 1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, -1, 0, 0

Offset: 1

T. D. Noe, Nov 03 2003

sign

Note that A049092 lists prime(n) such that a(n) = 0. Similarly, A078330 lists prime(n) such that a(n) = -1. See A088179 for prime(n) such that a(n) = 1. Also note that a(n) == A088144(n) (mod prime(n)).

J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 236.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Ed Pegg, Jr., Moebius Function (and squarefree numbers)
Eric Weisstein's World of Mathematics, Moebius Function

Cf. A089495 (mu(p+1) for prime p), A089496 (mu(p+1)+mu(p-1) for prime p), A089497 (mu(p+1)-mu(p-1) for prime p).

Cf. A049092, A078330, A088144, A088179.

[MoebiusMu(NthPrime(n)-1): n in [1..100]]; // Vincenzo Librandi, Dec 23 2018

Mathematica
```
Table[MoebiusMu[Prime[n]-1], {n, 150}]
```
PARI
```
a(n)=moebius(prime(n)-1)
```

a(n) = A067460(n) - 1. - Benoit Cloitre, Nov 04 2003

If p = prime(n), then a(n) is congruent modulo p to the sum of all primitive roots modulo p. [Uspensky and Heaslet]. - Michael Somos, Feb 16 2020

A088179 Primes p such that mu(p-1) = 1; that is, p-1 is squarefree and has an even number of prime factors, where mu is the Moebius function.

Original entry on oeis.org

2, 7, 11, 23, 47, 59, 83, 107, 167, 179, 211, 227, 263, 331, 347, 359, 383, 463, 467, 479, 503, 547, 563, 571, 587, 691, 719, 839, 859, 863, 887, 911, 967, 983, 1019, 1123, 1187, 1231, 1283, 1291, 1303, 1307, 1319, 1327, 1367, 1439, 1483, 1487, 1523, 1619, 1723

Offset: 1

N. J. A. Sloane and T. D. Noe, Nov 03 2003

nonn

It is an unsolved problem to determine if this sequence has a positive density in the primes. - Pieter Moree (moree(AT)mpim-bonn.mpg.de), Nov 03 2003

Except for the initial element 2, this sequence seems to be exactly those primes the sum of whose nonquadratic, nonprimitive-root residues is congruent to -1(mod p). - Dimitri Papadopoulos, Jan 10 2016

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Moebius Function

Cf. A049092 (primes p with mu(p-1)=0), A078330 (primes p with mu(p-1)=-1), A089451 (mu(p-1) for prime p).

Cf. A002496.

[n: n in [2..2000] | IsPrime(n) and MoebiusMu(n-1) eq 1]; // Vincenzo Librandi, Jan 10 2016

filter:= proc(p) isprime(p) and numtheory:-mobius(p-1) = 1 end proc:
select(filter, [2,seq(i,i=3..2000,2)]); # Robert Israel, Feb 03 2016

Select[Prime[Range[400]], MoebiusMu[ #-1]==1&]

lista(nn) = forprime(p=2, nn, if (moebius(p-1) == 1, print1(p, ", "))); \\ Michel Marcus, Jan 10 2016

list(lim)=my(v=List(),last); forsquarefree(k=1,lim\1, if(moebius(k)==1, last=k[1], if(k[2][,2]==[1]~ && k[1]-last==1, listput(v,k[1])))); Vec(v) \\ Charles R Greathouse IV, Jan 08 2018

A049092 Primes p such that p-1 is not squarefree.

Original entry on oeis.org

5, 13, 17, 19, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 127, 137, 149, 151, 157, 163, 173, 181, 193, 197, 199, 229, 233, 241, 251, 257, 269, 271, 277, 281, 293, 307, 313, 317, 337, 349, 353, 373, 379, 389, 397, 401, 409, 421, 433, 449, 457, 461, 487

Offset: 1

Labos Elemer

nonn

Primes p with mu(p-1)=0, where mu is the Möbius function. - T. D. Noe, Nov 03 2003
Primes p such that the sum of the primitive roots of p (see A088144) is 0 mod p. - Jon Wharf, Mar 12 2015
The relative density of this sequence within the primes is 1 - A005596 = 0.626044... - Amiram Eldar, Feb 10 2021

			p = 257 is here because p-1 = 256 = 2^8.
p = 997 is here because p-1 = 996 = 3*(2^2)*83.

T. D. Noe, Table of n, a(n) for n=1..1000
Eric Weisstein's World of Mathematics, Moebius Function.

Cf. A005596, A039787, A078330 (primes p with mu(p-1)=-1), A088179 (primes p such that mu(p-1)=1), A089451 (mu(p-1) for prime p), A145199.

[ p: p in PrimesUpTo(500) | not IsSquarefree(p-1) ]; // Vincenzo Librandi, Mar 12 2015

Select[Prime[Range[400]], MoebiusMu[ #-1]==0&]

forprime(p=2,500,if(!issquarefree(p-1),print(p))) \\ Michael B. Porter, Mar 16 2015

a(n) = A145199(n) + 1. - Amiram Eldar, Feb 10 2021

A244466 Nonprimes n such that mu(phi(n)) = 1.

Original entry on oeis.org

1, 9, 14, 18, 22, 46, 94, 118, 166, 214, 334, 358, 422, 454, 526, 662, 694, 718, 766, 926, 934, 958, 961, 1006, 1094, 1126, 1142, 1174, 1382, 1438, 1678, 1718, 1726, 1774, 1822, 1849, 1922, 1934, 1966, 2038, 2246, 2374, 2462, 2566, 2582, 2606, 2614, 2638, 2654, 2734, 2878, 2966, 2974, 3046

Offset: 1

Torlach Rush, Jun 28 2014

Odd terms > 1 are the square of some prime: a(2) = 9 = 3^2, a(23) = 961 = 31^2, a(36) = 1849 = 43^2, ... .

Odd terms > 1 are A078330^2. Even terms are 2*A088179 and 2*A078330^2. - Robert Israel, Aug 01 2014

			9 is not prime, phi(9) = 6 and mu(6) = 1, mu(phi(9)) = 1, so 9 is here.

Michael De Vlieger, Table of n, a(n) for n = 1..11320

Cf. A078330, A088179.

C
```
a(n) {return mu(phi(n))==1 ? n : ;}
```

filter:= n -> not isprime(n) and numtheory:-mobius(numtheory:-phi(n))=1:
select(filter, [$1..10000]); # Robert Israel, Aug 01 2014

Select[Range[3200],
And[MoebiusMu[EulerPhi[#]] == 1,
   Not[PrimeQ[#]]] &] (* Michael De Vlieger, Aug 06 2014 *)

for(n=1,10^4,if(moebius(eulerphi(n))==1,print1(n,", "))) \\ Derek Orr, Aug 01 2014

from sympy import totient,factorint,primefactors,isprime
[n for n in range(1,10**5) if n == 1 or (not isprime(n) and max(factorint(totient(n)).values()) < 2 and (-1)**len(primefactors(totient(n))) == 1)] # Chai Wah Wu, Aug 06 2014

A255199 Numbers k such that mu(k) = mu(phi(k)) where mu(k) is the Möbius function and phi(k) is Euler's totient function.

Original entry on oeis.org

1, 3, 8, 12, 14, 16, 20, 22, 24, 25, 27, 28, 31, 32, 36, 40, 43, 44, 45, 46, 48, 50, 52, 54, 56, 60, 63, 64, 67, 68, 71, 72, 75, 76, 79, 80, 81, 84, 88, 90, 92, 94, 96, 99, 100, 103, 104, 108, 112, 116, 117, 118, 120, 124, 125, 126, 128, 131, 132, 135, 136, 139

Offset: 1

Tom Edgar, Feb 16 2015

nonn

If k and phi(k) are both not squarefree then k is in the list.
A prime p is in the list if p - 1 is squarefree and bigomega(p - 1) = A001222(p - 1) is odd.
It follows that the subsequence of primes is A078330. - Bernard Schott, Apr 03 2021

			8 is in the list since mu(8) = 0 and mu(phi(8)) = mu(4) = 0.
7 is not in the list since mu(7) = -1 and mu(phi(7)) = mu(6) = 1.

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

Cf. A000010, A008683, A033631, A013929, A078330.

Select[Range[200], MoebiusMu[#] == MoebiusMu[EulerPhi[#]] &] (* Alonso del Arte, Feb 16 2015 *)

for(n=1, 140, if(moebius(n) == moebius(eulerphi(n)), print1(n,", "))) \\  Indranil Ghosh, Mar 11 2017

[n for n in [1..1000] if moebius(n)==moebius(euler_phi(n))]

A244723 Nonprimes n such that, mu(n) = mu(phi(n)).

Original entry on oeis.org

1, 8, 12, 14, 16, 20, 22, 24, 25, 27, 28, 32, 36, 40, 44, 45, 46, 48, 50, 52, 54, 56, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 88, 90, 92, 94, 96, 99, 100, 104, 108, 112, 116, 117, 118, 120, 124, 125, 126, 128, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 166

Offset: 1

Torlach Rush, Jul 04 2014

Odd terms, with the exception of 1, are not squarefree. - Torlach Rush, Jul 22 2014.

2*p, where p is an odd prime, is in the sequence iff p is in A088179. - Robert Israel, Jul 31 2014

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A078330, A000010, A008683, A053850, A088179.

filter:= proc(n) uses numtheory;  not(isprime(n)) and mobius(n) = mobius(phi(n)) end proc;
select(filter, [$1..1000]); # Robert Israel, Jul 31 2014

searchMax = 200; Complement[Select[Range[searchMax], MoebiusMu[#] == MoebiusMu[EulerPhi[#]] &], Prime[Range[PrimePi[searchMax]]]] (* Alonso del Arte, Jul 05 2014 *)
Select[Range[searchMax], !PrimeQ[#] && MoebiusMu[#] == MoebiusMu[EulerPhi[#]]& ] (* Zak Seidov, Jul 31 2014 *)

for(n=1,10^3,if(!isprime(n)&&moebius(eulerphi(n))==moebius(n),print1(n,", "))) \\ Derek Orr, Jul 30 2014

A255219 Squarefree numbers k such that mu(k) = mu(phi(k)) where mu(k) is the Möbius function and phi(k) is Euler's totient function.

Original entry on oeis.org

1, 3, 14, 22, 31, 43, 46, 67, 71, 79, 94, 103, 118, 131, 139, 166, 191, 214, 223, 239, 283, 311, 334, 358, 367, 419, 422, 431, 439, 443, 454, 499, 526, 599, 607, 619, 643, 647, 659, 662, 683, 694, 718, 743, 766, 787, 823, 827, 907, 926, 934, 947, 958, 971, 1006

Offset: 1

Tom Edgar, Feb 17 2015

nonn

A prime p is a term in the sequence if p - 1 is squarefree and bigomega(p - 1) = A001222(p - 1) is odd (see A078330).

			31 is a term since mu(31) = -1 and mu(phi(31)) = mu(30) = -1.
7 is not a term since mu(7) = -1 and mu(phi(7)) = mu(6) = 1.
24 is not a term since mu(24) = 0 (i.e., 24 is not squarefree).

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

Cf. A000010, A001222, A005117, A008683, A033631, A078330, A255199.

Select[Range[1000], Abs[MoebiusMu[#] + MoebiusMu[EulerPhi[#]]] == 2 &] (* Alonso del Arte, Feb 17 2015 *)

for(n=1, 1006, if(abs(moebius(n) + moebius(eulerphi(n))) == 2, print1(n,", "))) \\ Indranil Ghosh, Mar 10 2017

[n for n in [1..1006] if moebius(n)==moebius(euler_phi(n)) if moebius(n)!=0]

A096163 Primes p of the form qrs + 1 where q, r and s are distinct primes.

Original entry on oeis.org

31, 43, 67, 71, 79, 103, 131, 139, 191, 223, 239, 283, 311, 367, 419, 431, 439, 443, 499, 599, 607, 619, 643, 647, 659, 683, 743, 787, 823, 827, 907, 947, 971, 1031, 1039, 1087, 1091, 1103, 1163, 1223, 1259, 1399, 1427, 1447, 1499, 1511, 1543, 1559, 1571, 1579

Offset: 1

Rick L. Shepherd, Jun 18 2004

nonn

Each composite number qrs = a(n)-1 is a squarefree 3-almost prime. This sequence is a subsequence of A078330 which, besides having 3 as its first term, first differs by including 2311 = 2*3*5*7*11 + 1 (a squarefree 5-almost prime plus 1).

Cf. A078330 (primes p with mu(p-1) = -1).

With[{nn=50},Take[Union[Select[Times@@@Subsets[Prime[Range[2nn]],{3}]+1,PrimeQ]],nn]] (* Harvey P. Dale, Jun 06 2021 *)

/* Here are five equivalent PARI programs */ forprime(p=2,2400, if(moebius(p-1)==-1 && omega(p-1)==3, print1(p,","))) forprime(p=2,2400, if(moebius(p-1)==-1 && bigomega(p-1)==3, print1(p,","))) forprime(p=2,2400, if(bigomega(p-1)==3 && omega(p-1)==3, print1(p,","))) forprime(p=2,2400, if(omega(p-1)==3 && issquarefree(p-1), print1(p,","))) forprime(p=2,2400, if(bigomega(p-1)==3 && issquarefree(p-1), print1(p,",")))

A322665 Partial sums of A089451.

Original entry on oeis.org

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

Offset: 1

Jianing Song, Dec 22 2018

sign

a(n) is the number of primes p among the first n ones such that the sum of primitive roots is congruent to +1 modulo p minus the number of primes p among the first n ones such that the sum of primitive roots is congruent to -1 modulo p. Here, the prime number 2 is counted in the minuends but not in the subtrahends.
Although there are more positive terms among the first few ones, there are 5887 negative ones among the first 10000 terms, along with 237 zeros.
The largest terms among the first 10000 ones are a(n) = 41 for n in {8389, 8749, 8750, 8751, 8752, 8753}, and the smallest being a(n) = -41 for n in {4037, 4038, 4039, 4040, 4041, 4043, 4044, 4045, 4063, 4064, 4065, 4081, 4082, 4083, 4086, 4098, 4099, 4100}. What is the rate of growth for sup_{i=1..n} a(i) and inf_{i=1..n} a(i)?

			prime(11) = 31, mu(1) = mu(6) = mu(10) = mu(22) = +1, mu(2) = mu(30) = -1, so a(11) = 4 - 2 = 2.
prime(22) = 79, mu(1) = mu(6) = mu(10) = mu(22) = mu(46) = mu(58) = +1, mu(2) = mu(30) = mu(42) = mu(66) = mu(70) = mu(78) = -1, so a(22) = 6 - 6 = 0.

Cf. A089451.

Cf. Also A049092, A078330, A088144, A088179.

PARI
```
a(n) = sum(i=1, n, moebius(prime(i)-1))
```

cp's OEIS Frontend

A089451 a(n) = mu(prime(n)-1), where mu is the Moebius function (A008683).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A088179 Primes p such that mu(p-1) = 1; that is, p-1 is squarefree and has an even number of prime factors, where mu is the Moebius function.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

A049092 Primes p such that p-1 is not squarefree.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A244466 Nonprimes n such that mu(phi(n)) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

C

Maple

Mathematica

PARI

Python

A255199 Numbers k such that mu(k) = mu(phi(k)) where mu(k) is the Möbius function and phi(k) is Euler's totient function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Sage

A244723 Nonprimes n such that, mu(n) = mu(phi(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A255219 Squarefree numbers k such that mu(k) = mu(phi(k)) where mu(k) is the Möbius function and phi(k) is Euler's totient function.

Views

Author