A131181 -id:A131181 - cp's OEIS frontend

A000379 Numbers where total number of 1-bits in the exponents of their prime factorization is even; a 2-way classification of integers: complement of A000028.

1, 6, 8, 10, 12, 14, 15, 18, 20, 21, 22, 26, 27, 28, 32, 33, 34, 35, 36, 38, 39, 44, 45, 46, 48, 50, 51, 52, 55, 57, 58, 62, 63, 64, 65, 68, 69, 74, 75, 76, 77, 80, 82, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 100, 106, 111, 112, 115, 116, 117, 118, 119, 120, 122, 123, 124, 125, 129

Offset: 1

N. J. A. Sloane

This sequence and A000028 (its complement) give the unique solution to the problem of splitting the positive integers into two classes in such a way that products of pairs of distinct elements from either class occur with the same multiplicities [Lambek and Moser]. Cf. A000069, A001969.
See A000028 for precise definition, Maple program, etc.
The sequence contains products of even number of distinct terms of A050376. - Vladimir Shevelev, May 04 2010
From Vladimir Shevelev, Oct 28 2013: (Start)
Numbers m such that the infinitary Möbius function (A064179) of m equals 1. (This follows from the definition of A064179.)
A number m is in the sequence iff the number k = k(m) of terms of A050376 that divide m with odd maximal exponent is even (see example).
(End)
Numbers k for which A064547(k) [or equally, A268386(k)] is even. Numbers k for which A010060(A268387(k)) = 0. - Antti Karttunen, Feb 09 2016
The sequence is closed under the commutative binary operation A059897(.,.). As integers are self-inverse under A059897(.,.), it therefore forms a subgroup of the positive integers considered as a group under A059897(.,.). Specifically (expanding on the comment above dated May 04 2010) it is the subgroup of even length words in A050376, which is the group's lexicographically earliest ordered minimal set of generators. A000028, the set of odd length words in A050376, is its complementary coset. - Peter Munn, Nov 01 2019
From Amiram Eldar, Oct 02 2024: (Start)
Numbers whose number of infinitary divisors (A037445) is a square.
Numbers whose exponentially odious part (A367514) has an even number of distinct prime factors, i.e., numbers k such that A092248(A367514(k)) = 0. (End)

			If m = 120, then the maximal exponent of 2 that divides 120 is 3, for 3 it is 1, for 4 it is 1, for 5 it is 1. Thus k(120) = 4 and 120 is a term. - _Vladimir Shevelev_, Oct 28 2013

Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 22.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
J. Lambek and L. Moser, On some two way classifications of integers, Canad. Math. Bull. 2 (1959), 85-89.
Eric Weisstein's World of Mathematics, Group.
Wikipedia, Generating set of a group.

Subsequences: A030229, A238748, A262675, A268390.
Subsequence of A268388 (apart from the initial 1).
Complement: A000028.
Sequences used in definitions of this sequence: A133008, A050376, A059897, A064179, A064547, A124010 (prime exponents), A268386, A268387, A010060.
Other 2-way classifications: A000069/A001969 (to which A000120 and A010060 are relevant), A000201/A001950.
This is different from A123240 (e.g., does not contain 180). The first difference occurs already at n=31, where A123240(31) = 60, a value which does not occur here, as a(31+1) = 62. The same is true with respect to A131181, as A131181(31) = 60.
Cf. A030231, A037445, A092248, A367514.

a000379 n = a000379_list !! (n-1)
a000379_list = filter (even . sum . map a000120 . a124010_row) [1..]
-- Reinhard Zumkeller, Oct 05 2011

Select[ Range[130], EvenQ[ Count[ Flatten[ IntegerDigits[#, 2]& /@ Transpose[ FactorInteger[#]][[2]]], 1]]&] // Prepend[#, 1]& (* Jean-François Alcover, Apr 11 2013, after Harvey P. Dale *)

is(n)=my(f=factor(n)[,2]); sum(i=1,#f,hammingweight(f[i]))%2==0 \\ Charles R Greathouse IV, Aug 31 2013
(Scheme, two variants)
(define A000379 (MATCHING-POS 1 1 (COMPOSE even? A064547)))
(define A000379 (MATCHING-POS 1 1 (lambda (n) (even? (A000120 (A268387 n))))))
;; Both require also my IntSeq-library. - Antti Karttunen, Feb 09 2016

Edited by N. J. A. Sloane, Dec 20 2007, to restore the original definition.

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.

Original entry on oeis.org

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

Clark Kimberling

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

			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.

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

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.

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 *)

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

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

cp's OEIS Frontend

A000379 Numbers where total number of 1-bits in the exponents of their prime factorization is even; a 2-way classification of integers: complement of A000028.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Extensions