A008334 -id:A008334 - cp's OEIS frontend

A023508 Sum of exponents in prime-power factorization of n-th prime - 1.

0, 1, 2, 2, 2, 3, 4, 3, 2, 3, 3, 4, 4, 3, 2, 3, 2, 4, 3, 3, 5, 3, 2, 4, 6, 4, 3, 2, 5, 5, 4, 3, 4, 3, 3, 4, 4, 5, 2, 3, 2, 5, 3, 7, 4, 4, 4, 3, 2, 4, 4, 3, 6, 4, 8, 2, 3, 5, 4, 5, 3, 3, 4, 3, 5, 3, 4, 6, 2, 4, 6, 2, 3, 4, 5, 2, 3, 5, 6, 5, 3, 5, 3, 7, 3, 3, 7, 5, 4, 4, 2, 2, 6, 4, 3, 2, 3, 5, 4

Offset: 1

Clark Kimberling

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001222, A008334, A006093, A210936.

Array[Plus@@Last/@FactorInteger[Prime[ # ]-1]&,6! ] (* Vladimir Joseph Stephan Orlovsky, Feb 28 2010 *)
PrimeOmega[Prime[Range[100]] - 1] (* Amiram Eldar, Apr 16 2024 *)

a(n) = bigomega(prime(n)-1); \\ Amiram Eldar, Apr 16 2024

a(n) = A001222(A006093(n)). - R. J. Mathar, Feb 06 2019

A067466 Primes p such that p-1 has 2 distinct prime factors.

Original entry on oeis.org

7, 11, 13, 19, 23, 29, 37, 41, 47, 53, 59, 73, 83, 89, 97, 101, 107, 109, 113, 137, 149, 163, 167, 173, 179, 193, 197, 227, 233, 251, 263, 269, 293, 317, 347, 353, 359, 383, 389, 401, 433, 449, 467, 479, 487, 503, 509, 557, 563, 569, 577, 587, 593, 641, 653

Offset: 1

Benoit Cloitre, Feb 23 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006093, A007774, A008334.

Select[Prime[Range[120]], PrimeNu[#-1] == 2 &] (* Amiram Eldar, Jun 06 2022 *)

lista(nn) = {forprime(p=1, nn, if (omega(p-1) == 2, print1(p, ", ")););} \\ Michel Marcus, Nov 22 2013

A067467 Primes p such that p-1 has 3 distinct prime factors.

Original entry on oeis.org

31, 43, 61, 67, 71, 79, 103, 127, 131, 139, 151, 157, 181, 191, 199, 223, 229, 239, 241, 271, 277, 281, 283, 307, 311, 313, 337, 349, 367, 373, 379, 397, 409, 419, 431, 439, 443, 457, 461, 491, 499, 521, 523, 541, 599, 601, 607, 613, 617, 619, 643, 647, 659

Offset: 1

Benoit Cloitre, Feb 23 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006093, A008334.

Select[Prime[Range[150]],PrimeNu[#-1]==3&] (* Harvey P. Dale, Mar 13 2022 *)

lista(nn) = {forprime(p=1, nn, if (omega(p-1) == 3, print1(p, ", ")););} \\ Michel Marcus, Nov 22 2013

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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A023508 Sum of exponents in prime-power factorization of n-th prime - 1.

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A067466 Primes p such that p-1 has 2 distinct prime factors.

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PARI

A067467 Primes p such that p-1 has 3 distinct prime factors.

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PARI

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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