A049098 Primes p such that p+1 is divisible by a square.

3, 7, 11, 17, 19, 23, 31, 43, 47, 53, 59, 67, 71, 79, 83, 89, 97, 103, 107, 127, 131, 139, 149, 151, 163, 167, 179, 191, 197, 199, 211, 223, 227, 233, 239, 241, 251, 263, 269, 271, 283, 293, 307, 311, 331, 337, 347, 349, 359, 367, 379, 383, 419, 431, 439, 443

Offset: 1

Author

Labos Elemer

Keywords

• nonn
• easy
• nice

Comments

Numbers m such that A010051(m)*(1-A008966(m+1)) = 1. - Reinhard Zumkeller, May 21 2009

This sequence is infinite and its relative density in the sequence of primes is equal to 1 - Product_{p prime} (1-1/(p*(p-1))) = 1 - A005596 = 0.626044... (Mirsky, 1949). - Amiram Eldar, Feb 14 2021

Examples

			31 is a term because 32 is divisible by a square, 16.
101 is not a term because 102 = 2*3*17 is squarefree.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Leon Mirsky, The number of representations of an integer as the sum of a prime and a k-free integer, The American Mathematical Monthly, Vol. 56, No. 1 (1949), pp. 17-19.

Crossrefs

Cf. A005596, A008966, A010051, A049097 (complement with respect to A000040), A160696.

Programs

Haskell

a049098 n = a049098_list !! (n-1)
a049098_list = filter ((== 0) . a008966 . (+ 1)) a000040_list
-- Reinhard Zumkeller, Oct 18 2011

Maple

with(numtheory): a := proc (n) if isprime(n) = true and issqrfree(n+1) = false then n else end if end proc: seq(a(n), n = 1 .. 500); # Emeric Deutsch, Jun 21 2009

Mathematica

Select[Prime[Range[200]],!SquareFreeQ[#+1]&]   (* Harvey P. Dale, Mar 27 2011 *)
Select[Prime[Range[200]], MoebiusMu[# + 1] == 0 &] (* Alonso del Arte, Oct 18 2011 *)

PARI

forprime(p=2,1e4,if(!issquarefree(p+1),print1(p", "))) \\ Charles R Greathouse IV, Oct 18 2011

Formula

A160696(a(n)) > 1. - Reinhard Zumkeller, May 24 2009