A328137 -id:A328137 - cp's OEIS frontend

A328160 Terms k of A112998 such that k+2 is nonsquarefree.

61, 73, 277, 421, 2797, 6217, 8521, 9277, 9817, 10357, 11161, 12301, 12841, 13381, 15121, 17377, 17881, 18097, 19861, 25657, 30517, 30661, 33037, 35521, 36241, 36457, 48121, 50821, 51481, 54421, 56437, 58417, 60217, 66601, 66697, 67057, 71341, 74077, 77641, 79801, 88117, 94777, 96181, 98017

Offset: 1

Robert Israel, Oct 05 2019

nonn

Complement of A328137 in A112998.

Each term is either 3*x^2-2 where x, 3*x^2-2 and (3*x^2-1)/2 are prime or it is 9*x-2 where x, 9*x-2 and (9*x-1)/2 are prime.

			a(3)=277 is a term because 277 is prime, 277+1=2*139 where 139 is prime, and 279=3^2*31 is a 3-almost prime that is nonsquarefree.

Robert Israel, Table of n, a(n) for n = 1..7487

Cf. A112998, A328137.

N:= 100000:
A1:= map(x -> 3*x^2-2, select(x -> isprime(x) and isprime(3*x^2-2) and isprime((3*x^2-1)/2), {seq(i,i=3..floor(sqrt((N+2)/3)),2)})):
A2:= map(x -> 9*x-2, select(x -> isprime(x) and isprime(9*x-2) and isprime((9*x-1)/2), {seq(i,i=3..(N+2)/9,2)})):
sort(convert(A1 union A2,list));

Select[Prime@ Range[10^4], And[PrimeOmega /@ {# + 1, # + 2} == {2, 3}, ! SquareFreeQ[# + 2]] &] (* Michael De Vlieger, Oct 06 2019 *)

A376352 Squarefree semiprimes k such that k+1 is the product of three distinct primes and k+2 is the product of four distinct primes.

Original entry on oeis.org

2413, 6193, 6697, 9469, 11065, 11233, 11893, 12153, 13333, 13393, 14005, 14089, 14233, 15293, 17113, 17533, 17833, 17869, 18613, 18653, 19693, 20053, 20557, 20613, 20733, 20893, 20993, 21145, 22033, 22285, 22405, 22693, 22753, 22969, 23329, 23413, 24033, 24493, 26101, 26453, 27113, 27553, 28117, 28453, 28741, 29053, 29353, 29713

Offset: 1

Massimo Kofler, Sep 21 2024

nonn

			2413 is a term because 2413 = 19*127 is the product of two distinct primes, 2414 = 2*17*71 is the product of three distinct primes and 2415 = 3*5*7*23 is the product of four distinct primes.
6193 is a term because 6193 = 11*563 is the product of two distinct primes, 6194 = 2*19*163 is the product of three distinct primes and 6195 = 3*5*7*59 is the product of four distinct primes.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A006881, A007304, A046386, A328137.

q:= n-> andmap(j-> map(i-> i[2], ifactors(n+j-2)[2])=[1$j], [$2..4]):
select(q, [$1..30000])[];  # Alois P. Heinz, Sep 21 2024

Position[Partition[FactorInteger[#][[;; , 2]] & /@ Range[30000], 3, 1], {{1, 1}, {1, 1, 1}, {1, 1, 1, 1}}] // Flatten (* Amiram Eldar, Sep 21 2024 *)

a(n) == 1 (mod 4).

cp's OEIS Frontend

A328160 Terms k of A112998 such that k+2 is nonsquarefree.

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