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

Author

Olivier Gérard

Keywords

• nonn

Comments

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

Examples

			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
		

Links

• 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.

Crossrefs

Cf. A000010, A007947, A049092 (complement).

Programs

Magma

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

Maple

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

Mathematica

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

PARI

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

PARI

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

Extensions

More terms from Labos Elemer