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A154598 a(n) is the smallest prime p such that p-1 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

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

LINKS

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

EXAMPLE

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.

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-Fran├žois Alcover, Dec 08 2016 *)

PROG

(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

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

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Last modified June 17 00:17 EDT 2021. Contains 345080 sequences. (Running on oeis4.)