A049092 -id:A049092 - cp's OEIS frontend

A039787 Primes p such that p-1 is squarefree.

2, 3, 7, 11, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107, 131, 139, 167, 179, 191, 211, 223, 227, 239, 263, 283, 311, 331, 347, 359, 367, 383, 419, 431, 439, 443, 463, 467, 479, 499, 503, 547, 563, 571, 587, 599, 607, 619, 643, 647, 659, 683, 691, 719, 743

Offset: 1

Olivier Gérard

nonn

An equivalent definition: numbers n such that phi(n) is equal to the squarefree kernel of n-1.
Minimal value of first differences (between odd terms) is 4. - Zak Seidov, Apr 16 2013
The density of this set in A000040 is Artin's constant A = A005596 = 37.39...%, see Mirsky. - Charles R Greathouse IV, Oct 26 2015

			phi(43)=42, 42=2^1*3^1*7^1, 2*3*7=42.
p=223 is here because p-1=222=2*3*37

N. J. A. Sloane, Table of n, a(n) for n = 1..25000, Oct 25 2015 (extending earlier b-file of _Zak Seidov_)
Theodor Estermann, Einige Sätze über quadratfreie Zahlen, Math. Ann. 105:1 (1931), pp. 653-662.
Leon Mirsky, The number of representations of an integer as the sum of a prime and a k-free integer, American Mathematial Monthly 56:1 (1949), pp. 17-19.

Cf. A000010, A007947, A049092 (complement).

[p: p in PrimesUpTo(780) | IsSquarefree(p-1)];  // Bruno Berselli, Mar 03 2011

isA039787 := proc(n)
    if isprime(n) then
        numtheory[issqrfree](n-1) ;
    else
        false;
    end if;
end proc:
for n from 2 to 100 do
    if isA039787(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Apr 17 2013
with(numtheory): lis:=[]; for n from 1 to 10000 do if issqrfree(ithprime(n)-1) then lis:=[op(lis), ithprime(n)]; fi; od: lis; # N. J. A. Sloane, Oct 25 2015

Select[Prime[Range[132]],SquareFreeQ[#-1]&](* Zak Seidov, Aug 22 2012 *)

is(n)=isprime(n) && issquarefree(n-1) \\ Charles R Greathouse IV, Jul 02 2013

forprime(p=2, 1e3, if(issquarefree(p-1), print1(p", "))); \\ Altug Alkan, Oct 26 2015

More terms from Labos Elemer

A078330 Primes p such that mu(p-1) = -1, where mu is the Moebius function; that is, p-1 is squarefree and has an odd number of prime factors.

Original entry on oeis.org

3, 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

Offset: 1

Shyam Sunder Gupta, Nov 21 2002

			31 is in the sequence because 31 is a prime and mu(30) = -1.
37 is not in the sequence because, although 37 is prime, mu(36) = 0.

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

Cf. A008683, A049092 (primes p with mu(p-1) = 0), A088179 (primes p with mu(p-1) = 1), A089451 (mu(p-1) for prime p).

Select[Prime[Range[400]], MoebiusMu[# - 1] == -1 &] (* from T. D. Noe *)

j=[]; forprime(n=1,2000,if(moebius(n)==moebius(n-1),j=concat(j,n))); j

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

Original entry on oeis.org

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

A145199 Nonsquarefree numbers k such that k+1 is prime.

Original entry on oeis.org

4, 12, 16, 18, 28, 36, 40, 52, 60, 72, 88, 96, 100, 108, 112, 126, 136, 148, 150, 156, 162, 172, 180, 192, 196, 198, 228, 232, 240, 250, 256, 268, 270, 276, 280, 292, 306, 312, 316, 336, 348, 352, 372, 378, 388, 396, 400, 408, 420, 432, 448, 456, 460, 486, 490

Offset: 1

Giovanni Teofilatto, Oct 04 2008

			4 is in the sequence because it is not squarefree and 5 is prime. - _Emeric Deutsch_, Oct 12 2008

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A006093, A013929, A049092, A077064. Includes A152680.

[n: n in [1..5*10^2]| not IsSquarefree(n) and IsPrime(n+1)]; // Vincenzo Librandi, Dec 24 2015

with(numtheory): a:=proc(n) if issqrfree(n)=false and isprime(n+1)=true then n else end if end proc: seq(a(n),n=1..600); # Emeric Deutsch, Oct 12 2008
with(numtheory): a:=proc(k) if issqrfree(ithprime(k)-1)=false then ithprime(k)-1 else end if end proc: seq(a(k),k=1..110); # Emeric Deutsch, Oct 12 2008

Select[Prime[Range[120]]-1, !SquareFreeQ[ # ]&] (* T. D. Noe, Oct 06 2008 *)

is(n)=isprime(n+1) && !issquarefree(n) \\ Charles R Greathouse IV, Jun 13 2017

a(n) = A049092(n) - 1. - Amiram Eldar, Feb 10 2021

Corrected and extended by T. D. Noe, Emeric Deutsch and R. J. Mathar, Oct 05 2008

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

A039787 Primes p such that p-1 is squarefree.

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A078330 Primes p such that mu(p-1) = -1, where mu is the Moebius function; that is, p-1 is squarefree and has an odd number of prime factors.

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

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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.

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A145199 Nonsquarefree numbers k such that k+1 is prime.

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A322665 Partial sums of A089451.

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