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

1,1

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.

Links

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

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}.

Example

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.

Mathematica

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

Prog

(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
(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

Crossrefs

Cf. A001358, A056809, A070552.

Sequence in context: A130021 A003864 A182602 * A053750 A321439 A271921

Adjacent sequences: A123252 A123253 A123254 * A123256 A123257 A123258

Keyword

easy,nonn

Author

Jonathan Vos Post, Oct 09 2006

Extensions

336 and 680 added by Vincenzo Librandi, Dec 22 2010

Status

approved