A154598 a(n) is the smallest prime p such that p-1 and p+1 both have n prime factors (with multiplicity).

5, 19, 89, 271, 1889, 10529, 75329, 157951, 3885569, 11350529, 98690561, 65071999, 652963841, 6548416001, 253401579521, 160283668481, 1851643543553, 3450998226943, 23114453401601, 1194899749142527, 1101483715526657, 7093521158963201

Offset: 2

Author

J. M. Bergot, Jan 12 2009

Keywords

• nonn
• nice
• hard

Comments

Factors are counted with multiplicity. Sequence begins at a(2) since no prime p exists such that the adjacent numbers p-1 and p+1 have just one factor. For p = 2, p-1 has zero factors; for p >= 3, p+1 has at least two factors.

a(24) > 2^54. - Jon E. Schoenfield, Feb 08 2009

Examples

			For p = 19, p-1 = 18 = 2*3*3 and p+1 = 20 = 2*2*5 both have three factors and 19 is the smallest such prime. For p = 271, p-1 = 270 = 2*3*3*3*5 and p+1 = 272 = 2*2*2*2*17 both have five factors and 271 is the smallest prime surrounded by numbers with five factors.
For p = 89, p-1 = 88 = 2*2*2*11 and p+1 = 90 = 2*3*3*5 both have four factors and 89 is the smallest such prime. For p = 1889, p-1 = 1888 = 2*2*2*2*2*59 and p+1 = 1890 = 2*3*3*3*5*7 both have six factors and 1889 is the smallest prime surrounded by numbers with six factors.
		

Crossrefs

Cf. A001222 (number of prime divisors of n).

Programs

Mathematica

a[2] = 5; a[n_] := a[n] = For[p = NextPrime[.63(*empirical*)*a[n-1]], True, p = NextPrime[p], If[PrimeOmega[p-1] == n && PrimeOmega[p+1] == n, Return[p]]]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 2, 14}] (* Jean-François Alcover, Dec 08 2016 *)

PARI

{for(n=2, 14, p=2; while(!(bigomega(p-1)==n&&bigomega(p+1)==n), p=nextprime(p+1)); print1(p, ","))}

PARI

a(n)=forprime(p=2,, if(bigomega(p-1)==n && bigomega(p+1)==n, return(p))) \\ Charles R Greathouse IV, Apr 27 2015

Extensions

Edited, 2 removed, 151 replaced by 89 and a(6) - a(14) added by Klaus Brockhaus, Jan 12 2009

a(15) from Klaus Brockhaus, Jan 14 2009

a(16)-a(20) from Jon E. Schoenfield and Donovan Johnson, Jan 21 2009

a(21) from Jon E. Schoenfield, Jan 27 2009

a(22) from Jon E. Schoenfield, Jan 28 2009

a(23) from Jon E. Schoenfield, Jan 30 2009