

A154598


a(n) is the smallest prime p such that p1 and p+1 both have n prime factors (with multiplicity).


4



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

2,1


COMMENTS

Factors are counted with multiplicity. Sequence begins at a(2) since no prime p exists such that the adjacent numbers p1 and p+1 have just one factor. For p = 2, p1 has zero factors; for p >= 3, p+1 has at least two factors.
a(24) > 2^54.  Jon E. Schoenfield, Feb 08 2009


LINKS

Table of n, a(n) for n=2..23.


EXAMPLE

For p = 19, p1 = 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, p1 = 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, p1 = 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, p1 = 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.


MATHEMATICA

a[2] = 5; a[n_] := a[n] = For[p = NextPrime[.63(*empirical*)*a[n1]], True, p = NextPrime[p], If[PrimeOmega[p1] == n && PrimeOmega[p+1] == n, Return[p]]]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 2, 14}] (* JeanFrançois Alcover, Dec 08 2016 *)


PROG

(PARI) {for(n=2, 14, p=2; while(!(bigomega(p1)==n&&bigomega(p+1)==n), p=nextprime(p+1)); print1(p, ", "))}
(PARI) a(n)=forprime(p=2, , if(bigomega(p1)==n && bigomega(p+1)==n, return(p))) \\ Charles R Greathouse IV, Apr 27 2015


CROSSREFS

Cf. A001222 (number of prime divisors of n).
Sequence in context: A149800 A147099 A323788 * A184513 A149801 A149802
Adjacent sequences: A154595 A154596 A154597 * A154599 A154600 A154601


KEYWORD

nonn,nice,hard


AUTHOR

J. M. Bergot, Jan 12 2009


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


STATUS

approved



