A155800 -id:A155800 - cp's OEIS frontend

A239669 Total number of prime factors counted with multiplicity of prime(n)-1 and prime(n)+1, where prime(n) is the n-th prime.

1, 3, 4, 5, 5, 5, 7, 6, 6, 6, 8, 6, 7, 6, 7, 7, 6, 6, 6, 8, 7, 8, 6, 8, 9, 7, 7, 7, 8, 8, 11, 7, 7, 7, 7, 8, 6, 8, 7, 6, 7, 8, 10, 9, 8, 9, 7, 9, 6, 7, 8, 9, 9, 9, 11, 7, 8, 10, 6, 8, 6, 7, 8, 8, 7, 6, 7, 9, 6, 8, 9, 8, 8, 7, 9, 10, 7, 7, 9, 8, 8, 7, 10, 10

Offset: 1

Chris Boyd, Mar 23 2014

nonn

If a(n) <= 6, n is a distinguished prime (A106639). According to a conjecture in A106639, 6 is the least term appearing infinitely often. Numerical evidence suggests the conjecture that other - perhaps all - integers > 6 appear infinitely often. The strong conjecture implies that all terms but one in A155800 are positive.

			a(7) = 7 as prime(7) = 17, 17-1 = 2^4 and 17+1 = 2*3^2, so total number of prime factors = 4+3 = 7.

Chris Boyd, Table of n, a(n) for n = 1..10000

Cf. A000040, A106639, A155800.

forprime(p=1,500,print1(bigomega(p-1) + bigomega(p+1)", "))

a(n) = Omega(prime(n)^2 - 1), where Omega is A001222.

A155850 a(n) = smallest k > 1 such that k-1 and k+1 together have n prime divisors.

Original entry on oeis.org

2, 4, 3, 5, 7, 15, 17, 31, 65, 129, 127, 449, 511, 2561, 1025, 7937, 12799, 20481, 8191, 28673, 65537, 131071, 458751, 360449, 966655, 524287, 4194303, 2097151, 29360129, 34865153, 67108865, 134217729, 33554431, 608174081, 268435457, 536870911, 4831838207

Offset: 1

Klaus Brockhaus, Jan 28 2009, Jan 31 2009

nonn

Prime divisors are counted with multiplicity.
Similar to A155800, where k is restricted to primes.
Terms of the form 2^m-1 or 2^m+1 seem to occur frequently.

			Adjacent to 2 are the numbers 1 and 3 which together have one prime divisor, hence a(1) = 2. Adjacent to 3 are 2 and 4; together they have three prime divisors, hence a(3) = 3. Adjacent to 4 are the primes 3 and 5, each having one prime divisor; hence a(2) = 4.
For k = 129, k-1 = 128 = 2*2*2*2*2*2*2 and k+1 = 130 = 2*5*13 together have ten prime divisors. For all numbers k < 129 the adjacent numbers k-1 and k+1 together have fewer or more than ten (for 127 there are eleven) prime divisors, hence a(10) = 129.

Cf. A001222 (number of prime divisors of n), A155800, A154704, A154598.

{for(n=2, 150000000, s=bigomega(n-1)+bigomega(n+1); if(v[s]==0, v[s]=n)); v}

a(34)-a(37) from Donovan Johnson, Nov 02 2013

cp's OEIS Frontend

A239669 Total number of prime factors counted with multiplicity of prime(n)-1 and prime(n)+1, where prime(n) is the n-th prime.

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