A280383 -id:A280383 - cp's OEIS frontend

A280382 Numbers k such that k-1 has the same number of prime factors counted with multiplicity as k+1.

4, 5, 6, 12, 18, 19, 29, 30, 34, 42, 43, 50, 51, 55, 56, 60, 67, 69, 72, 77, 86, 89, 92, 94, 102, 108, 115, 120, 122, 138, 142, 144, 150, 151, 160, 171, 173, 180, 184, 186, 187, 189, 192, 197, 198, 202, 204, 214, 216, 218, 220, 228, 233, 236, 237, 240, 243, 245, 248, 249, 266, 267, 270, 271, 274, 282

Offset: 1

Rick L. Shepherd, Jan 01 2017

nonn

			Unlike for A088070, 5 is a term here because 4 = 2^2 and 6 = 2*3 each have two prime factors when counted with multiplicity. Similarly, 3 is not a term of this sequence (but is in A088070) because 2 and 4 have different numbers of prime factors as counted by A001222.

Rick L. Shepherd, Table of n, a(n) for n = 1..10000

Cf. A001222, A088070 (similar but prime factors counted without multiplicity), A280383 (prime factor count is same both ways), A280469 (subsequence of current with k-1 and k+1 squarefree also), A045920 (similar but for k and k+1).

Cf. A115167 (subsequence of odd terms).

Select[Range[2, 300], Equal @@ PrimeOmega[# + {-1, 1}] &] (* Amiram Eldar, May 20 2021 *)

IsInA280382(n) = n > 1 && bigomega(n-1) == bigomega(n+1)

from sympy import primeomega
def aupto(limit):
  prv, cur, nxt, alst = 1, 1, 2, []
  for n in range(3, limit+1):
    if prv == nxt: alst.append(n)
    prv, cur, nxt = cur, nxt, primeomega(n+2)
  return alst
print(aupto(282)) # Michael S. Branicky, May 20 2021

A280469 Numbers n such that n-1 and n+1 are squarefree and have the same number of prime factors.

Original entry on oeis.org

4, 6, 12, 18, 30, 34, 42, 56, 60, 72, 86, 92, 94, 102, 108, 138, 142, 144, 150, 160, 180, 184, 186, 192, 198, 202, 204, 214, 216, 218, 220, 228, 236, 240, 248, 266, 270, 282, 300, 302, 304, 312, 320, 322, 328, 340, 348, 392, 394, 412, 414, 416, 420, 432, 446

Offset: 1

Rick L. Shepherd, Jan 03 2017

nonn

For a given term n of this sequence, n-1 and n+1 are both squarefree k-almost primes for the same k. The sequence is thus the union of the averages (arithmetic means) of twin prime pairs (A014574), the averages of twin squarefree semiprime pairs, the averages of twin squarefree 3-almost prime pairs, ... (where "twin ... pairs" means the members of each pair differ by two). A subsequence of A280382 and of A280383.

			The number 34 is a term because 33 = 3*11 and 35 = 5*7, a twin semiprime pair. Unlike A280382 and A280383, 19 is not a term here because 18 = 2*3^2 and 20  = 2^2*5, neither of which is squarefree.

Rick L. Shepherd, Table of n, a(n) for n = 1..10000

Cf. A014574, A280382, A280383.

Select[Range@ 500, And[Times @@ First@ # == 1, SameQ @@ Last@ #] &@ Transpose@ Map[{Boole@ SquareFreeQ@ #, PrimeNu@ #} &, # + {-1, 1}] &] (* Michael De Vlieger, Jan 30 2017 *)

IsInA280469(n) = n > 1 && issquarefree(n-1) && issquarefree(n+1) && omega(n-1) == omega(n+1)

cp's OEIS Frontend

A280382 Numbers k such that k-1 has the same number of prime factors counted with multiplicity as k+1.

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