A179384 -id:A179384 - cp's OEIS frontend

A378458 Squarefree numbers k such that k + 1 is squarefree but k + 2 is not.

2, 6, 10, 14, 22, 30, 34, 38, 42, 46, 58, 61, 66, 70, 73, 78, 82, 86, 94, 102, 106, 110, 114, 118, 122, 130, 133, 138, 142, 145, 154, 158, 166, 173, 178, 182, 186, 190, 194, 202, 205, 210, 214, 218, 222, 226, 230, 238, 246, 254, 258, 262, 266, 273, 277, 282

Offset: 1

Gus Wiseman, Dec 02 2024

nonn

These are the positions of 2 in A378369 (difference between n and the next nonsquarefree number).

The asymptotic density of this sequence is Product_{p prime} (1 - 2/p^2) - Product_{p prime} (1 - 3/p^2) = A065474 - A206256 = 0.19714711803343537224... . - Amiram Eldar, Dec 03 2024

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

Complement of A007675 within A007674.
The version for prime power instead of nonsquarefree is a subset of A006549.
Another variation is A073247.
The version for nonprime instead of squarefree is A179384.
Positions of 0 in A378369 are A013929.
Positions of 1 in A378369 are A373415.
Positions of 2 in A378369 are A378458 (this).
Positions of 3 in A378369 are A007675.
A000961 lists the powers of primes, differences A057820.
A120327 gives the least nonsquarefree number >= n.
A378373 counts composite numbers between nonsquarefree numbers.
Cf. A065474, A067535, A081221, A206256, A236575, A377046, A378033, A378039, A378086.

Select[Range[100],NestWhile[#+1&,#,SquareFreeQ[#]&]==#+2&]

list(lim) = my(q1 = 1, q2 = 1, q3); for(k = 3, lim, q3 = issquarefree(k); if(q1 && q2 &&!q3, print1(k-2, ", ")); q1 = q2; q2 = q3); \\ Amiram Eldar, Dec 03 2024

A221865 The nonprimes k such that k + 2 is either a prime or a semiprime.

Original entry on oeis.org

0, 1, 4, 8, 9, 12, 15, 20, 21, 24, 27, 32, 33, 35, 36, 39, 44, 45, 49, 51, 55, 56, 57, 60, 63, 65, 69, 72, 75, 77, 80, 81, 84, 85, 87, 91, 92, 93, 95, 99, 104, 105, 111, 116, 117, 119, 120, 121, 125, 129, 132, 135, 140, 141, 143, 144, 147, 153, 155, 156, 159, 161, 164, 165, 171, 175, 176, 177, 183

Offset: 1

Gerasimov Sergey, Apr 18 2013

nonn

Chen primes A109611(n) such that A109611(n)-/+ a(n) are both prime: 2, 29, 53, 113, 139,...
Unrelated: Numbers n such that n + 2^bigomega(n) is either a prime or a semiprime: 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 15, 17, 18, 19, 21, 22, 23, 25, 27,...
A179384 is a subsequence. - R. J. Mathar, Apr 26 2013

Cf. A001358, A109611, A175634.

A221865 := proc(n)
    option remember;
    if n =1 then
        0;
    else
        for a from procname(n-1)+1 do
            if not isprime(a) then
            if isprime(a+2) or numtheory[bigomega](a+2) = 2 then
                return a;
            end if;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Apr 26 2013

Select[Range[0,200],!PrimeQ[#]&&PrimeOmega[#+2]<3&] (* Harvey P. Dale, May 05 2013 *)

Corrected by R. J. Mathar, Apr 26 2013

cp's OEIS Frontend

A378458 Squarefree numbers k such that k + 1 is squarefree but k + 2 is not.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI