A026416 A 2-way classification of integers: a(1) = 1, a(2) = 2 and for n > 2, a(n) is the smallest number not of the form a(i)*a(j) for 1 <= i < j < n.

1, 2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, 23, 24, 25, 29, 30, 31, 37, 40, 41, 42, 43, 47, 49, 53, 54, 56, 59, 61, 66, 67, 70, 71, 73, 78, 79, 81, 83, 88, 89, 97, 101, 102, 103, 104, 105, 107, 109, 110, 113, 114, 121, 127, 128, 130, 131, 135, 136, 137, 138, 139

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

An equivalent definition is: a(1) = 1, a(2) = 2; and for n > 2, a(n) = least positive integer > a(n-1) and not of the form a(i)*a(j) for 1 <= i < j < n.

a(2) to a(29) match the initial terms of A000028. [corrected by Peter Munn, Mar 15 2019]

This has a simpler definition than A000028, but the resulting pair lacks the crucial property of the A000028/A000379 pair (see the comment in A000028). - N. J. A. Sloane, Sep 28 2007

Contains (for example) 180, so is different from A123193. - Max Alekseyev, Sep 20 2007

From Vladimir Shevelev, Apr 05 2013: (Start)

1) The sequence does not contain (for example) 140, so is different from A000028.

2) Representation of numbers which are absent in the sequence as a product of two different terms of the sequence is, generally speaking, not unique. For example, 210 = 2*105 = 3*70 = 5*42 = 7*30.

(End)

Excluding a(1) = 1, the lexicographically earliest sequence of distinct nonnegative integers such that no term is a product of 2 distinct terms. Removing the latter distinctness requirement, the sequence becomes A026424; and the equivalent sequence where the product is of 2 or more distinct terms is A050376. A000028 is similarly the equivalent sequence when A059897 is used as multiplicative operator in place of standard integer multiplication. - Peter Munn, Mar 15 2019

Examples

			a(8) is not 10 because we already have 10 = 2*5. Of course all primes appear. 16 appears because 16 is not a product of earlier terms.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..400 from Vincenzo Librandi)

Crossrefs

Complement of A131181. Cf. A000028, A059897.

Cf. A066724, A026477, A050376, A084400.

Similar sequences with different starting conditions: A026417 (1,3), A026419 (1,4), A026420 (2,4), A026421 (3,4).

Related sequences with definition using any products (not necessarily distinct) and with various starting conditions: A026422 (1,2),A026423 (1,3), A026424 (2,3), A026425 (1,4), A026426 (2,4), A026427 (3,4).

See also families of related sequences: A026431 (excluding product-1), A026443 (excluding product+2), A026453 (excluding product-2) and references therein.

Programs

Mathematica

a[1]=1; a[2]=2; a[n_] := a[n] = For[k = a[n-1] + 1, True, k++, If[ FreeQ[ Table[ a[i]*a[j], {i, 1, n-2}, {j, i+1, n-1}], k], Return[k]]]; Table[a[n], {n, 1, 101}] (* Jean-François Alcover, May 16 2013 *)

Python

from itertools import count, islice
def agen(): # generator of terms
    a, products = [1, 2], {2}
    yield from a
    for k in count(3):
        if k not in products:
            yield k
            products.update(k*a[i] for i in range(len(a)))
            a.append(k)
        products.discard(k)
print(list(islice(agen(), 62))) # Michael S. Branicky, Jun 09 2025

Extensions

More terms from Max Alekseyev, Sep 23 2007

Edited by N. J. A. Sloane, Jul 13 2008 at the suggestion of R. J. Mathar and Max Alekseyev