A135257 Smallest semiprimes such that a(j) - a(k) are all different.

4, 6, 9, 10, 21, 34, 55, 65, 74, 94, 141, 177, 203, 219, 298, 321, 395, 403, 529, 581, 635, 662, 731, 934, 982, 1073, 1145, 1317, 1418, 1762, 1769, 1849, 1915, 2066, 2165, 2305, 2327, 2746, 2809, 3127, 3409, 3859, 3983, 4138, 4285, 4559, 4742, 5186, 5299, 5854

Offset: 1

Author

Jonathan Vos Post, Dec 01 2007

Keywords

• nonn

Comments

Semiprime analog of A079848. This is the slowest-growing semiprime B2 sequence (see A005282). The difference set through a(n) must have cardinality exactly C(n,2) = n*(n+1)/2 = A000217(n-1).

Examples

			a(1) = 4, the smallest semiprime (A001358(1)).
a(2) = 6 = A001358(2); the difference set so far is {2}.
a(3) = 9 = A001358(3); the difference set so far is {2,3,5}.
a(4) = 10 = A001358(4); the difference set so far is {1,2,3,4,5,6}.
a(5) can't be 14 = A001358(5) because 14-10 =4, in the difference set through a(4); can't be 15 = A001358(6) because 15-10 =5, in the difference set through a(4)...
a(5) = 21 = A001358(7); the difference set so far is {1,2,3,4,5,6,11,12,15,17}.
a(6) = 34 = A001358(12); the difference set so far is {1,2,3,4,5,6,11,12,13,15,17,24,25,28,30}.
		

Crossrefs

Cf. A000217, A001358, A005282, A079848.

Programs

Maple

A135257 := proc(nmax) local a,anext,diffset,good,an ; a := [4,6] ; diffset := {2} ; while nops(a) < nmax do for anext from op(-1,a)+1 do if numtheory[bigomega](anext) = 2 then good := true ; for an in a do if ( anext-an) in diffset then good := false ; break ; fi ; od; if good then for an from 1 to nops(a) do diffset := diffset union { anext-op(an,a) } ; od; a := [op(a),anext] ; break ; fi ; fi ; od ; od: a ; end: A135257(60) ; # R. J. Mathar, Jan 08 2008

Python

from itertools import count, islice
from sympy import factorint
def A135257_gen(): # generator of terms
    aset2, alist = set(), []
    for k in count(0):
        if sum(factorint(k).values()) == 2:
            bset2 = set()
            for a in alist:
                if (b:=k-a) in aset2:
                    break
                bset2.add(b)
            else:
                yield k
                alist.append(k)
                aset2.update(bset2)
A135257_list = list(islice(A135257_gen(),30)) # Chai Wah Wu, Sep 11 2023

Extensions

Corrected and extended by R. J. Mathar, Jan 08 2008