A123375 Least k such that the difference between consecutive semiprimes A065516(k) equals n, or 0 if no such k exists.

3, 1, 2, 4, 24, 6, 10, 56, 50, 78, 34, 320, 249, 186, 463, 762, 598, 1238, 422, 760, 3760, 3585, 9214, 1765, 4112, 13447, 6675, 4585, 68498, 8112, 10083, 8650, 86203, 49433, 35085, 20641, 458421, 8861, 366314, 157857, 169147, 487115, 277440, 563951, 511757, 920602, 75150

Offset: 1

Author

Alexander Adamchuk, Nov 09 2006

Keywords

• nonn
• hard

Comments

a(n) equals least k such that A065516(k) = n, or 0 if no such k exists. Conjecture: a(n)>0 exists for all n.

Examples

			A065516(n) begins {2, 3, 1, 4, 1, 6, 1, 3, 1, 7, 1, 1, 3, 1, 7, 3, 2, 4, 2, 1, 4, 3, 4, 5, ...}.
Thus a(1) = 3, a(2) = 1, a(3) = 2, a(4) = 4, a(5) = 24.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..82
• Eric Weisstein's World of Mathematics, Semiprime

Crossrefs

Cf. A001358 (semiprimes), A065516 (differences between semiprimes), A131109.

Programs

Maple

nextSprime := proc(n) local res ; res := n+1 ; while numtheory[bigomega](res) <> 2 do res := res+1 ; od ; RETURN(res) ; end ; nmax := 500 ; kmax := 500000 ; a := array(1..nmax) ; for i from 1 to nmax do a[i] := 0 ; od : sp1 := 4 : sp2 := nextSprime(sp1) : n := sp2-sp1 : a[n] := 1 : for k from 2 to kmax do sp1 := sp2 ; sp2 := nextSprime(sp1) ; n := sp2-sp1 ; if a[n] = 0 then a[n] := k ; fi ; od : for i from 1 to nmax do if a[i] = 0 then break ; else printf("%d,",a[i]) ; fi ; od : # R. J. Mathar, Jan 13 2007

Mathematica

Table[k=6;While[FreeQ[b=Differences[Select[Range@k++,PrimeOmega[#]==2&]],n]];
Length@b,{n,11}] (* Giorgos Kalogeropoulos, Apr 02 2021 *)

Extensions

Corrected and extended by R. J. Mathar, Jan 13 2007

More terms from David A. Corneth, Apr 02 2021