A260360 - OEIS

%I #27 Aug 08 2021 13:55:38

%S 0,1,2,2,1,1,8,4,2,2,2,2,16,10,16,24,6,4,4,10,28,30,8,2,12,36,50,4,0,

%T 6,4,6,14,32,8,10,80,40,46,84,14,16,4,4,4,30,76,94,10,12,12,0,3,129,

%U 64,62,18,16,40,26,56,14,18,66,68,4,166,144,18,168,118,30,24,184,94,86,6,12,2,12,36,40,70,56,10

%N The absolute difference between the largest prime factors of prime(n)-1 and prime(n+1)-1.

%C a(n)=0 if and only if n is in A105403.

%C It is an open question whether there are infinitely many zeros in this sequence. Are there infinitely many terms below some fixed upper bound?

%H Robert Israel, <a href="/A260360/b260360.txt">Table of n, a(n) for n = 2..10000</a>

%F a(n) = abs(A023503(n+1) - A023503(n)). - _Robert Israel_, Aug 06 2015

%e n=4: The prime factors of prime(4)-1 are 2,3 and the prime factors of prime(5)-1 are 2,5. The largest are 3 and 5, so a(4)=2.

%p B:= [seq(max(numtheory:-factorset(ithprime(i)-1)),i=2..101)]:

%p seq(abs(B[n+1]-B[n]),n=1..99); # _Robert Israel_, Aug 06 2015

%t Table[Abs[FactorInteger[Prime[n] - 1][[-1, 1]] - FactorInteger[Prime[n + 1] - 1][[-1, 1]]], {n, 2, 86}] (* _Michael De Vlieger_, Jul 24 2015 *)

%t Rest[Abs[Differences[Table[FactorInteger[p-1][[-1,1]],{p,Prime[ Range[ 90]]}]]]] (* _Harvey P. Dale_, Aug 08 2021 *)

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

%o a(n) = gpf(prime(n)-1) - gpf(prime(n+1)-1); \\ _Michel Marcus_, Aug 05 2015

%Y Cf. A023503, A105403.

%K nonn

%O 2,3

%A _Pratik Koirala_, _Otis Tweneboah_, _Nathan Fox_, _Eugene Fiorini_, Jul 23 2015