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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.

%I A026416 #48 Jun 09 2025 14:41:03
%S A026416 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,
%T A026416 53,54,56,59,61,66,67,70,71,73,78,79,81,83,88,89,97,101,102,103,104,
%U A026416 105,107,109,110,113,114,121,127,128,130,131,135,136,137,138,139
%N 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.
%C A026416 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.
%C A026416 a(2) to a(29) match the initial terms of A000028. [corrected by _Peter Munn_, Mar 15 2019]
%C A026416 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
%C A026416 Contains (for example) 180, so is different from A123193. - _Max Alekseyev_, Sep 20 2007
%C A026416 From _Vladimir Shevelev_, Apr 05 2013: (Start)
%C A026416 1) The sequence does not contain (for example) 140, so is different from A000028.
%C A026416 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.
%C A026416 (End)
%C A026416 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
%H A026416 Michael S. Branicky, <a href="/A026416/b026416.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..400 from Vincenzo Librandi)
%e A026416 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.
%t A026416 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 *)
%o A026416 (Python)
%o A026416 from itertools import count, islice
%o A026416 def agen(): # generator of terms
%o A026416     a, products = [1, 2], {2}
%o A026416     yield from a
%o A026416     for k in count(3):
%o A026416         if k not in products:
%o A026416             yield k
%o A026416             products.update(k*a[i] for i in range(len(a)))
%o A026416             a.append(k)
%o A026416         products.discard(k)
%o A026416 print(list(islice(agen(), 62))) # _Michael S. Branicky_, Jun 09 2025
%Y A026416 Complement of A131181. Cf. A000028, A059897.
%Y A026416 Cf. A066724, A026477, A050376, A084400.
%Y A026416 Similar sequences with different starting conditions: A026417 (1,3), A026419 (1,4), A026420 (2,4), A026421 (3,4).
%Y A026416 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).
%Y A026416 See also families of related sequences: A026431 (excluding product-1), A026443 (excluding product+2), A026453 (excluding product-2) and references therein.
%K A026416 nonn,easy
%O A026416 1,2
%A A026416 _Clark Kimberling_
%E A026416 More terms from _Max Alekseyev_, Sep 23 2007
%E A026416 Edited by _N. J. A. Sloane_, Jul 13 2008 at the suggestion of _R. J. Mathar_ and _Max Alekseyev_