A259562 Numbers n such that the sum of the distinct prime factors of prime(n)-1 and prime(n+1)-1 are the same.

2, 414, 556, 3962, 4972, 6151, 6521, 8440, 8665, 13769, 13909, 15576, 16696, 17176, 19926, 20630, 21541, 27090, 30822, 62118, 65349, 74014, 94203, 98600, 101231, 103058, 108333, 112332, 136036, 142714, 145588, 147150, 160730, 162366, 169137, 194681, 200837

Offset: 1

Author

Otis Tweneboah, Pratik Koirala, Eugene Fiorini, Nathan Fox, Jun 30 2015

Keywords

• nonn

Comments

Although there are more terms than A105403 so far, these numbers are still fairly uncommon.

Is this sequence infinite?

It would follow from the generalized Bunyakovsky conjecture that, e.g., there are infinitely many primes p such that p+2, p+12, p+14, 6*p^2+84*p+1 and 6*p^2+84*p+145 are all prime, and there are no primes between 6*p^2+84*p+1 and 6*p^2+84*p+145. If so, then the sequence is infinite, because it contains n where prime(n) = 6*p^2+84*p+1, with prime(n)-1 having distinct prime factors 2,3,p,p+14 and prime(n+1) having distinct prime factors 2,3,p+2,p+12. - Robert Israel, Jun 30 2015

Examples

			The prime factors of prime(414)-1 are 2,3,5,5,19 and the prime factors of prime(415)-1 are 2,2,2,3,7,17. The sum of the distinct entries in each of these lists is 29.
		

Crossrefs

Cf. A008472, A105403, A259564.

Programs

Maple

Primes:= select(isprime,[2,seq(2*i+1,i=1..10^6)]):
spf:= map(p -> convert(numtheory:-factorset(p-1),`+`), Primes):
select(t -> spf[t+1]=spf[t], [$1..nops(Primes)-1]);

Mathematica

Select[Range@ 250000, Total[First /@ FactorInteger[Prime@ # - 1]] == Total[First /@ FactorInteger[Prime[# + 1] - 1]] &] (* Michael De Vlieger, Jul 01 2015 *)

PARI

spf(n) = {my(f=factor(n)); sum(k=1, #f~, f[k,1]);}
lista(nn) = {forprime(p=2, nn, if (spf(p-1)==spf(nextprime(p+1)-1), print1(primepi(p), ", ")););} \\ Michel Marcus, Jun 30 2015

Extensions

More terms from Alois P. Heinz, Jun 30 2015