A105403 Numbers n such that prime(n)-1 and prime(n+1)-1 have the same largest prime factor.

2, 30, 53, 217

Offset: 1

Author

Cino Hilliard, May 01 2005

Keywords

• hard
• nonn

Comments

The PARI program below is a generalization of this type of sequence. These numbers are rare. Are they finite? Proof?

No more terms between 217 and 3090000. - R. J. Mathar, Feb 14 2008

No more terms < 5760000. - David Wasserman, May 02 2008

No more terms < 11078936. - Otis Tweneboah, Pratik Koirala, Eugene Fiorini, Nathan Fox, Jun 30 2015

No more terms < 50000000 (~40 minutes computation time). - Manfred Scheucher, Jul 25 2015

No more terms < 100000000. Conjecture: sequence is full. - Jon E. Schoenfield, Jul 25 2015

Examples

			The prime factorization of prime(217) - 1 = 1327 - 1 = 1326 is 2*3*13*17, and that of prime(218) - 1 = 1361 - 1 = 1360 is 2^4*5*17; each has 17 as its largest factor.
		

Links

• Manfred Scheucher, Sage Script

Crossrefs

Cf. A023503.

Programs

Mathematica

Select[Range@ 1000000, FactorInteger[Prime[#] - 1][[-1, 1]] == FactorInteger[Prime[# + 1] - 1][[-1, 1]] &] (* Michael De Vlieger, Jul 25 2015 *)

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) }

PARI

gpf(n)=if(n>1, my(f=factor(n)[,1]); f[#f], 1)
is(n,p=prime(n))=my(q=nextprime(p+1),g=gcd(p-1,q-1)); q\=g; p\=g; forprime(r=2,gpf(g), p/=r^valuation(p,r); q/=r^valuation(q,r)); p==1 && q==1
n=0;forprime(p=2,1e9,n++;if(is(0,p),print1(n", "))) \\ Charles R Greathouse IV, Aug 27 2015

Formula

{i: A023503(i) = A023503(i+1)}. - R. J. Mathar, Feb 14 2008