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A155800
a(n) = smallest prime p such that p-1 and p+1 together have n prime divisors, or a(n) = 0 if no such prime exists.
3
2, 0, 3, 5, 7, 19, 17, 31, 97, 191, 127, 449, 769, 3329, 6143, 7937, 12799, 51199, 8191, 165887, 65537, 131071, 1179649, 2654209, 7995391, 524287, 10616831, 12910591, 167772161, 113246209, 169869311, 155189249, 1887436799, 3221225473
OFFSET
1,1
COMMENTS
Prime divisors are counted with multiplicity.
LINKS
Donovan Johnson and Giovanni Resta, Table of n, a(n) for n = 1..46 (terms < 10^13, first 40 terms from Donovan Johnson)
EXAMPLE
For p=2, the adjacent numbers 1 and 3 together have one prime divisor, hence a(1) = 2. For p=3, the adjacent numbers 2 and 4 together have three prime divisors, hence a(3) = 3. For primes greater than 3, the adjacent numbers are composite and therefore together have at least four prime divisors, so no prime exists whose neighbors together have two prime divisors. Hence a(2) = 0.
For p = 19, p-1 = 18 = 2*3*3 and p+1 = 20 = 2*2*5 together have six prime divisors. All smaller primes are surrounded by numbers which together have fewer or more than six (for 17 there are seven) prime divisors, hence a(6) = 19.
MATHEMATICA
Join[{2, 0}, With[{m=SortBy[{PrimeOmega[#-1]+PrimeOmega[#+1], #}&/@Prime[ Range[200000]], First]}, Transpose[Flatten[Table[Select[m, #[[1]]==n&, 1], {n, 3, 24}], 1]][[2]]]] (* Harvey P. Dale, Sep 24 2013 *)
PROG
(PARI) N=10^7; default(primelimit, N); M=26;
/* M must be determined empirically for each N. Program-generated zeros other than a(2) appearing as terms when N and M are increased must be regarded as provisional */
for(m=1, M, t=0; forprime(p=1, N, if(bigomega(p-1)+bigomega(p+1)==m, t=1; print1(p", "); break)); if(t==0, print1("0, "))) \\ Chris Boyd, Mar 24 2014
CROSSREFS
Cf. A001222 (number of prime divisors of n), A154598, A155850, A154704.
Cf. comments in A239669.
Sequence in context: A216440 A094720 A334495 * A276658 A079510 A216255
KEYWORD
nonn
AUTHOR
J. M. Bergot, Jan 27 2009
EXTENSIONS
Edited, 1151 replaced by 769, 3457 replaced by 3329, extended beyond a(14) by Klaus Brockhaus, Jan 28 2009
a(29)-a(32) from Klaus Brockhaus, Jan 30 2009
a(33)-a(34) from Donovan Johnson, Aug 03 2009
STATUS
approved