A180868 Numbers n such that n and n+1 are semiprime powers.

9, 14, 15, 21, 25, 33, 34, 35, 38, 57, 64, 81, 85, 86, 93, 94, 118, 121, 122, 133, 141, 142, 145, 158, 177, 201, 202, 205, 213, 214, 215, 216, 217, 218, 225, 253, 298, 301, 302, 326, 334, 361, 381, 393, 394, 445, 446, 453, 481, 484, 501

Offset: 1

Author

Jonathan Vos Post, Jan 22 2011

Keywords

• nonn
• easy

Comments

This is to semiprimes A001358 and powers of semiprimes A085155 as A006549 is to primes A000040 and powers of primes A000961.

Examples

			15 is in the sequence because 15 = (3*5)^1 and 15+1 = 16 = (2*2)^2 are both semiprime powers.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000961, A001358, A006549, A085155.

Programs

Maple

spp:= proc(n) option remember; local l;
        if n<2 or isprime(n) then false
        else l:= ifactors(n)[2];
             if nops(l)>2 then false
           elif nops(l)=2 then evalb(l[1][2]=l[2][2])
           else evalb(irem(l[1][2], 2)=0)
             fi
        fi
      end:
a:= proc(n) option remember; local k;
      for k from 1+ `if`(n=1, 8, a(n-1))
        while not spp(k) or not spp(k+1)
      do od; k
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Jan 22 2011

Mathematica

sppQ[n_] := With[{f = FactorInteger[n][[All, 2]]}, n==1 || Length[f]==1 && EvenQ[f[[1]]] || Length[f]==2 && f[[1]]==f[[2]]];
Select[Range[1000], sppQ[#] && sppQ[#+1]&] (* Jean-François Alcover, Nov 21 2020 *)

Formula

{ n : {n,n+1} is subset of {A085155} } = { n : n = A001358(i)^j and n+1 = A001358(k)^m }.

Extensions

More terms and edited by Alois P. Heinz, Jan 22 2011