A008335 -id:A008335 - cp's OEIS frontend

A270992 Number of distinct prime divisors of prime(n)+1 and prime(n+1)+1.

0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1

Offset: 1

Michel Marcus, Mar 28 2016

nonn

			For n=1, p=2 and q=3; 3 and 4 have no common prime divisor, so a(1)=0.
For n=2, p=3 and q=5; 4 and 6 have 1 common prime divisor, so a(2)=1.
For n=9, p=23 and q=29; 24 and 30 have 2 common prime divisors, so a(9)=2.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A008335, A270592 (records).

Table[Length[Map[First, FactorInteger[GCD @@ {Prime@ n + 1, Prime[n + 1] + 1}]] /. 1 -> Nothing], {n, 101}] (* Michael De Vlieger, Mar 28 2016 *)
Length[Intersection[FactorInteger[#[[1]]+1][[All,1]],FactorInteger[#[[2]] + 1][[All,1]]]]&/@Partition[Prime[Range[120]],2,1] (* Harvey P. Dale, Jun 11 2017 *)

lista(nn) = {p = 2; f = factor(p+1)[,1]~; forprime(q=3, nn, g = factor(q+1)[,1]~; print1(#setintersect(f, g), ", "); p = q; f = g;);}

a(n) = my(p = prime(n), q = nextprime(p+1)); #setintersect(factor(p+1)[,1]~, factor(q+1)[,1]~);

A378123 a(n) = number of prime divisors of the sum of the first n odd primes.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 3, 2, 2, 2, 1, 3, 1, 2, 2, 3, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 3, 2, 3, 2, 2, 3, 3, 2, 3, 3, 3, 2, 4, 2, 2, 2, 3, 4, 3, 2, 3, 1, 3, 1, 3, 2, 3, 2, 5, 1, 5, 2, 4, 1, 3, 2, 3, 3, 3, 1, 3, 2, 3, 1, 3, 2, 3, 2, 4, 3, 3, 2, 2, 2, 4

Offset: 1

Clark Kimberling, Nov 17 2024

nonn

			3+5+7+11+13 = 39 = 3*13, so a(5) = 2.

Cf. A000040, A071215, A008334, A008335, A378122.

Cf. A001221, A071148.

a[n_] := PrimeNu[Total[Prime[1+Range[n]]]]; Array[a, 500]

a378123(n)=omega(sum(k=2,n+1,prime(k))) \\ Hugo Pfoertner, Nov 19 2024

a(n) = A001221(A071148(n)).

A245908 The number of distinct prime factors of prime(n)^2-1.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 2, 3, 3, 4, 3, 3, 4, 4, 3, 3, 4, 4, 4, 4, 3, 4, 4, 4, 3, 4, 4, 3, 4, 4, 3, 5, 4, 5, 4, 4, 4, 3, 4, 4, 4, 5, 4, 3, 4, 4, 5, 4, 4, 5, 4, 5, 4, 4, 3, 4, 4, 4, 4, 5, 4, 4, 5, 5, 4, 4, 5, 4, 4, 5, 4, 4, 4, 5, 5, 3, 5, 4, 4, 5

Offset: 1

R. J. Mathar, Aug 05 2014

nonn

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A008334, A008335, A082863.

[#PrimeDivisors(NthPrime(n)^2 -1): n in [1..100]]; // Vincenzo Librandi, Apr 27 2019

A245908 := proc(n)
    A082863(ithprime(n)) ;
end proc:

Table[PrimeNu[Prime[n]^2 - 1], {n, 100}] (* Wesley Ivan Hurt, Aug 05 2014 *)

vector(100, n, omega(prime(n)^2-1)) \\ Derek Orr, Aug 05 2014

a(n) = A082863(prime(n)).

a(n) = A008334(n) + A008335(n) - 1, if n>1.

A378122 a(n) = number of prime divisors of the sum of the first n primes.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 3, 3, 2, 4, 2, 2, 3, 4, 2, 2, 1, 3, 2, 3, 1, 2, 2, 3, 3, 3, 3, 3, 2, 3, 2, 2, 3, 4, 2, 3, 3, 2, 2, 2, 4, 3, 3

Offset: 1

Clark Kimberling, Nov 17 2024

nonn

			2+3+5+7+11 = 28 = 2*2*7, so a(5) = 2.

Cf. A000040, A071215, A008334, A008335, A378123.

a[n_] := PrimeNu[Total[Prime[Range[n]]]]; Array[a, 500]

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A270992 Number of distinct prime divisors of prime(n)+1 and prime(n+1)+1.

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A378123 a(n) = number of prime divisors of the sum of the first n odd primes.

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A245908 The number of distinct prime factors of prime(n)^2-1.

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A378122 a(n) = number of prime divisors of the sum of the first n primes.

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