A124936 Numbers k such that k - 1 and k + 1 are semiprimes.

5, 34, 50, 56, 86, 92, 94, 120, 122, 142, 144, 160, 184, 186, 202, 204, 214, 216, 218, 220, 236, 248, 266, 288, 290, 300, 302, 304, 320, 322, 328, 340, 392, 394, 412, 414, 416, 446, 452, 470, 472, 516, 518, 528, 534, 536, 544, 552, 580, 582, 590, 634, 668

Offset: 1

Author

Zak Seidov, Nov 13 2006

Keywords

• nonn

Comments

All but the first term are even.

Links

• Zak Seidov, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A092207 (k and k+2 are semiprimes), A086005 (k-1, k, k+1 are semiprimes), A086006 (primes p such that 2*p-1 and 2*p+1 are semiprimes), A082130 (2*k-1 and 2*k+1 are semiprimes).

Programs

Magma

IsSemiprime:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ n: n in [1..700] | IsSemiprime(n+1) and IsSemiprime(n-1)]; // Vincenzo Librandi, Mar 30 2015

Mathematica

lst={};Do[If[Plus@@Last/@FactorInteger[n-1]==2&&Plus@@Last/@FactorInteger[n+1]==2,AppendTo[lst,n]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 01 2009 *)
Select[Range[2, 700], PrimeOmega[# + 1] == PrimeOmega[# - 1] == 2 &] (* Vincenzo Librandi, Mar 30 2015 *)

PARI

list(lim)=if(lim<5,return([])); my(v=List([5]),x=1,y=1); forfactored(z=7,lim\1+1, if(vecsum(z[2][,2])==2 && vecsum(x[2][,2])==2, listput(v,z[1]-1)); x=y; y=z); Vec(v) \\ Charles R Greathouse IV, May 22 2018

Python

from sympy import factorint
from itertools import count, islice
def agen(): # generator of terms
    yield 5
    nxt = 0
    for k in count(6, 2):
        prv, nxt = nxt, sum(factorint(k+1).values())
        if prv == nxt == 2: yield k
print(list(islice(agen(), 53))) # Michael S. Branicky, Nov 26 2022

Formula

a(n) = A092207(n) + 1; at n>=2, a(n) = 2*A082130(n-1).