A105403 - OEIS

%I #35 Aug 27 2015 23:39:44

%S 2,30,53,217

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

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

%C No more terms between 217 and 3090000. - _R. J. Mathar_, Feb 14 2008

%C No more terms < 5760000. - _David Wasserman_, May 02 2008

%C No more terms < 11078936. - _Otis Tweneboah_, _Pratik Koirala_, _Eugene Fiorini_, _Nathan Fox_, Jun 30 2015

%C No more terms < 50000000 (~40 minutes computation time). - _Manfred Scheucher_, Jul 25 2015

%C No more terms < 100000000. Conjecture: sequence is full. - _Jon E. Schoenfield_, Jul 25 2015

%H Manfred Scheucher, <a href="/A105403/a105403.sage.txt">Sage Script</a>

%F {i: A023503(i) = A023503(i+1)}. - _R. J. Mathar_, Feb 14 2008

%e 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.

%t Select[Range@ 1000000, FactorInteger[Prime[#] - 1][[-1, 1]] == FactorInteger[Prime[# + 1] - 1][[-1, 1]] &] (* _Michael De Vlieger_, Jul 25 2015 *)

%o (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) }

%o (PARI) gpf(n)=if(n>1, my(f=factor(n)[,1]); f[#f], 1)

%o 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

%o n=0;forprime(p=2,1e9,n++;if(is(0,p),print1(n", "))) \\ _Charles R Greathouse IV_, Aug 27 2015

%Y Cf. A023503.

%K hard,nonn

%O 1,1

%A _Cino Hilliard_, May 01 2005