A285015 Primes of the form k * b^b - 1, with b > 1.

3, 7, 11, 19, 23, 31, 43, 47, 53, 59, 67, 71, 79, 83, 103, 107, 127, 131, 139, 151, 163, 167, 179, 191, 199, 211, 223, 227, 239, 251, 263, 269, 271, 283, 307, 311, 331, 347, 359, 367, 379, 383, 419, 431, 439, 443, 463, 467, 479, 487, 491, 499, 503, 523, 547

Offset: 1

Author

Vincenzo Librandi, May 08 2017

Keywords

• nonn
• easy

Comments

This sequence has relative density 1 - Prod_p 1-1/(p^p-p) = 0.52098749404893... in the primes, hence a(n) ~ kn log n where k = 1.91943... is the reciprocal of this quantity. - Charles R Greathouse IV, May 12 2017

Examples

			a(1) = 1*(2^2)-1 = 3.
a(2) = 2*(2^2)-1 = 7.
a(9) = 2*(3^3)-1 = 53.
		

Links

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

Crossrefs

Cf. A175768.

Programs

Maple

N:= 1000: # to get all terms <= N
bmax:= floor(ln(N+1)/LambertW(ln(N+1))):
sort(convert(select(isprime, {seq(seq(k*b^b-1, k=1..(N+1)/b^b),b=2..bmax)}),list)); # Robert Israel, May 11 2017

Mathematica

Take[Select[Union@Flatten@Table[k b^b - 1, {b, 2, 20}, {k, 148}], PrimeQ], 55]

PARI

upto(n)=my(l=List([3]), b=2, s=1); n++; while(b^b < n, c = b^b; forstep(i=2, n\c, s, if(isprime(i*c-1), listput(l, i*c-1))); s=3-s; b++); listsort(l, 1); l \\ David A. Corneth, May 11 2017

PARI

is(n)=if(!isprime(n), return(0)); my(t); forprime(p=2,, t=p^p; if((n+1)%t==0, return(1)); if(t>=n, return(0))) \\ Charles R Greathouse IV, May 11 2017