A176486 Numbers n such that semiprime(n)/prime(k)=prime and semiprime(n+1)/prime(k+1)=prime.

1, 2, 3, 4, 5, 6, 7, 10, 14, 18, 21, 29, 35, 36, 39, 41, 42, 45, 52, 58, 59, 62, 71, 73, 87, 91, 96, 97, 104, 116, 120, 127, 137, 141, 142, 156, 168, 169, 170, 178, 179, 181, 185, 188, 204, 211, 227, 245, 246, 249, 250, 254, 255, 261, 263, 279, 281, 285, 290, 297, 305

Offset: 1

Author

Juri-Stepan Gerasimov, Apr 18 2010

Keywords

• nonn

Comments

Indices n such that the (n+1)st semiprime has a prime factor which is the next prime after one of the prime factors of the n-th semiprime. - R. J. Mathar, Apr 20 2010

Examples

			a(1)=1 because semiprime(1)/prime(1)=2 and semiprime(2)/prime(2)=2;
a(2)=2 because semiprime(2)/prime(1)=3 and semiprime(3)/prime(2)=3;
a(3)=3 because semiprime(3)/prime(2)=3 and semiprime(4)/prime(3)=2.
		

Links

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

Crossrefs

Cf. A084126, A084127.

Programs

Maple

From R. J. Mathar, Apr 20 2010: (Start)
isA001358 := proc(n) numtheory[bigomega](n) = 2 ; end proc:
A001358 := proc(n) option remember ; if n = 1 then return 4 ; else for a from procname(n-1)+1 do if isA001358(a) then return a; end if; end do; end if; end proc:
A084126 := proc(n) min(op(numtheory[factorset](A001358(n)))) ; end proc:
A084127 := proc(n) max(op(numtheory[factorset](A001358(n)))) ; end proc:
A176486 := proc(n) if n = 1 then 1; else for a from procname(n-1)+1 do spl := A084126(a) ; sph := A084127(a) ; sp2l := A084126(a+1) ; sp2h := A084127(a+1) ; if sp2l = nextprime(spl) or sp2h = nextprime(spl) or sp2l = nextprime(sph) or sp2h = nextprime(sph) then return a; end if; end do: end if; end proc:
seq(A176486(n),n=1..80) ; (End)

Mathematica

sppQ[{a_,b_}]:=Module[{t=NextPrime[Transpose[FactorInteger[a]][[1]]],c,d}, c=t[[1]];d=If[Length[t]>1,t[[2]],t[[1]]];Divisible[b,c]|| Divisible[ b,d]]; Flatten[ Position[Partition[Select[Range[1500],PrimeOmega[#] == 2&],2,1],?sppQ]] (* _Harvey P. Dale, Mar 16 2015 *)

Extensions

Corrected (59, 137, 142 inserted, 147 removed) and extended by R. J. Mathar, Apr 20 2010