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A154598
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a(n) is the smallest prime p such that p-1 and p+1 both have n prime factors (with multiplicity).
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4
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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
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2,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A001222 (number of prime divisors of n).
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KEYWORD
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nonn,nice,hard
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AUTHOR
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EXTENSIONS
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Edited, 2 removed, 151 replaced by 89 and a(6) - a(14) added by Klaus Brockhaus, Jan 12 2009
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STATUS
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approved
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