A260360 The absolute difference between the largest prime factors of prime(n)-1 and prime(n+1)-1.

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, 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, 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

Offset: 2

Author

Pratik Koirala, Otis Tweneboah, Nathan Fox, Eugene Fiorini, Jul 23 2015

Keywords

• nonn

Comments

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

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

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A023503, A105403.

Programs

Maple

B:= [seq(max(numtheory:-factorset(ithprime(i)-1)),i=2..101)]:
seq(abs(B[n+1]-B[n]),n=1..99); # Robert Israel, Aug 06 2015

Mathematica

Table[Abs[FactorInteger[Prime[n] - 1][[-1, 1]] - FactorInteger[Prime[n + 1] - 1][[-1, 1]]], {n, 2, 86}] (* Michael De Vlieger, Jul 24 2015 *)
Rest[Abs[Differences[Table[FactorInteger[p-1][[-1,1]],{p,Prime[ Range[ 90]]}]]]] (* Harvey P. Dale, Aug 08 2021 *)

PARI

gpf(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
a(n) = gpf(prime(n)-1) - gpf(prime(n+1)-1); \\ Michel Marcus, Aug 05 2015

Formula

a(n) = abs(A023503(n+1) - A023503(n)). - Robert Israel, Aug 06 2015