A123255 Numbers k such that 4k+1, 4k+2, and 4k+3 are all semiprimes.

8, 21, 23, 30, 35, 50, 53, 54, 75, 98, 111, 158, 174, 210, 230, 260, 284, 315, 336, 350, 410, 440, 459, 473, 485, 495, 525, 545, 554, 576, 590, 608, 615, 629, 660, 680, 683, 774, 846, 900, 923, 966, 975, 989, 1071, 1103, 1133, 1148, 1220, 1400, 1430, 1463, 1499

Offset: 1

Author

Jonathan Vos Post, Oct 09 2006

Keywords

• easy
• nonn

Comments

4k+4 = 4*(k+1) = 2*2*(k+1) cannot be semiprime as well, as it has at least 3 prime factors with multiplicity. Thus there are no four consecutive semiprimes.

Examples

			a(1) = 8 because 4*8+1 = 33 = 3*11 is semiprime and 4*8+2 = 34 = 2*17 is semiprime and 4*8+3 = 35 = 3*5 is semiprime.
		

Links

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

Crossrefs

Cf. A001358, A056809, A070552.

Programs

Magma

IsSemiprime:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ n: n in [2..1500] | IsSemiprime(4*n+1) and IsSemiprime(4*n+2) and IsSemiprime(4*n+3) ]; // Vincenzo Librandi, Dec 22 2010

Mathematica

Select[Range[1100],Union[PrimeOmega[4#+{1,2,3}]]=={2}&] (* Harvey P. Dale, Feb 02 2015 *)

Python

from sympy import factorint, isprime
def issemiprime(n):
    return sum(factorint(n).values()) == 2 if n&1 else isprime(n//2)
def ok(n): return all(issemiprime(4*n+i) for i in (2, 1, 3))
print([k for k in range(1500) if ok(k)]) # Michael S. Branicky, Nov 26 2022

Formula

{k: 4k+1 is in A001358 AND 4k+2 is in A001358 AND 4k+3 is in A001358}.

{k: 4k+1 is in A070552 AND 4k+2 is in A070552}.

{(A056809(i)-1)/4}.

Extensions

336 and 680 added by Vincenzo Librandi, Dec 22 2010