A105308 Indices n of primes p(n), p(n+2) such that p(n)-1 and p(n+2)-1 have the same largest prime factor.

4, 6, 11, 45, 1408, 13313, 41752, 142122836, 48792948403

Offset

2,1

Comments

These numbers are rare. Are they finite? Proof?

a(10) > 1.2*10^12, if it exists. - Giovanni Resta, May 14 2016

Links

Table of n, a(n) for n=2..10.

Example

The prime factors of prime(45) - 1 = 2, 2, 7, 7;
the prime factors of prime(47) - 1 = 2, 3, 5, 7;
and 7 is the common largest factor.

Mathematica

t = {0, 0, 0}; Do[ t = {t[[2]], t[[3]], FactorInteger[ Prime[n + 2] - 1][[ -1, 1]]}; If[ t[[1]] == t[[3]], Print[n]], {n, 195000000}] (* Robert G. Wilson v, Jun 04 2005 *)

Prog

(PARI) /* prime indices such that gd of prime(x)+ k and prime(x+m) + k are equal */ divpm1(n, m, k) = { local(x, l1, l2, v1, v2); for(x=2, n, v1 = ifactor(prime(x)+ k); v2 = ifactor(prime(x+m)+k); l1 = length(v1); l2 = length(v2); if(v1[l1] == v2[l2], print1(x", ") ) ) }
ifactor(n) = /* Vector of the prime factors of n*/ { local(f, j, k, flist); flist=[]; f=Vec(factor(n)); for(j=1, length(f[1]), for(k = 1, f[2][j], flist = concat(flist, f[1][j]) ); ); return(flist) }

Crossrefs

Cf. A105404.

Sequence in context: A290651 A358913 A066155 * A116983 A196271 A078426

Adjacent sequences: A105305 A105306 A105307 * A105309 A105310 A105311

Keyword

more,nonn

Author

Cino Hilliard, May 01 2005

Extensions

a(8) from Robert G. Wilson v, Jun 04 2005

a(9) from Giovanni Resta, May 14 2016

Status

approved