A242489 Smallest even k such that lpf(k-1) = prime(n), while lpf(k-3) > prime(n), where lpf=least prime factor (A020639).

10, 26, 50, 254, 170, 392, 362, 944, 842, 1892, 1370, 2420, 1850, 2210, 3764, 6314, 3722, 4892, 5042, 7082, 8612, 9380, 7922, 12320, 11414, 10610, 11450, 13844, 18872, 16130, 17162, 20414, 19322, 26672, 24614, 25592, 29504, 37910, 29930, 44930, 36020, 36482

Offset: 2

Author

Vladimir Shevelev, May 16 2014

Keywords

• nonn

Comments

This sequence is connected with a sufficient condition for the infinitude of twin primes.

Almost all numbers of the form a(n)-3 are primes. For composite numbers of such a form, see A242716.

Primes p for which a(p) = p^2+1 form sequence A062326 for p >= 3. - Vladimir Shevelev, May 21 2014

Examples

			Let n=2, prime(2)=3. Then lpf(10-1)=3, but lpf(10-3)=7>3.
Since k=10 is the smallest such k, then a(2)=10.
		

Links

• Peter J. C. Moses, Table of n, a(n) for n = 2..2501

Crossrefs

Cf. A001359, A006512.

Programs

Mathematica

lpf[n_]:=lpf[n]=First[Select[Divisors[n],PrimeQ[#]&]];
Table[test=Prime[n];NestWhile[#+2&,test^2+1,!((lpf[#-1]==test)&&(lpf[#-3]>test))&],{n,2,60}] (* Peter J. C. Moses, May 21 2014 *)

PARI

a(n) = {k = 6; p = prime(n); while ((factor(k-1)[1, 1] != p) || (factor(k-3)[1, 1] <= p), k+= 2); k;} \\ Michel Marcus, May 16 2014

Formula

a(n) >= prime(n)^2+1. - Vladimir Shevelev, May 21 2014

Extensions

More terms from Michel Marcus, May 16 2014