A133008 -id:A133008 - 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.

A000028 Let k = p_1^e_1 p_2^e_2 p_3^e_3 ... be the prime factorization of n. Sequence gives k such that the sum of the numbers of 1's in the binary expansions of e_1, e_2, e_3, ... is odd.

Original entry on oeis.org

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, 60, 61, 66, 67, 70, 71, 72, 73, 78, 79, 81, 83, 84, 88, 89, 90, 96, 97, 101, 102, 103, 104, 105, 107, 108, 109, 110, 113, 114, 121, 126, 127, 128, 130, 131, 132, 135, 136, 137

Offset: 1

N. J. A. Sloane, Simon Plouffe

This sequence and A000379 (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.
Contains (for example) 180, so is different from A123193. - Max Alekseyev, Sep 20 2007
The sequence contains products of odd number of distinct terms of A050376. - Vladimir Shevelev, May 04 2010
From Vladimir Shevelev, Oct 28 2013: (Start)
Numbers m such that infinitary Moebius function of m (A064179) equals -1. This follows from the definition of A064179.
Number m is in the sequence if and only if the number k = k(m) of terms of A050376 which divide m with odd maximal exponent is odd.
For example, if m = 96, then the maximal exponent of 2 that divides 96 is 5, for 3 it is 1, for 4 it is 2, for 16 it is 1. Thus k(96) = 3 and 96 is a term.
(End)
Positions of odd terms in A064547, A268386 and A293439. - Antti Karttunen, Nov 09 2017
Lexicographically earliest sequence of distinct nonnegative integers such that no term is the A059897 product of 2 terms. (A059897 can be considered as a multiplicative operator related to the Fermi-Dirac factorization of numbers described in A050376.) Specifying that the A059897 product be of 2 distinct terms leaves the sequence unchanged. The equivalent sequences using standard integer multiplication are A026416 (with the 2 terms specified as distinct) and A026424 (otherwise). - Peter Munn, Mar 16 2019
From Amiram Eldar, Oct 02 2024: (Start)
Numbers whose number of infinitary divisors (A037445) is not a square.
Numbers whose exponentially odious part (A367514) has an odd number of distinct prime factors, i.e., numbers k such that A092248(A367514(k)) = 1. (End)

			If k = 96 then the maximal exponent of 2 that divides 96 is 5, for 3 it is 1. 5 in binary is 101_2 and has so has a sum of binary digits of 1 + 0 + 1 = 2. 1 in binary is 1_2 and so has a sum of binary digits of 1. Thus the sum of digits of binary exponents is 2 + 1 = 3 which is odd and so 96 is a term. - _Vladimir Shevelev_, Oct 28 2013, edited by _David A. Corneth_, Mar 20 2019

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.
Index entries for sequences computed from exponents in factorization of n.

Cf. A133008, A000379 (complement), A000120 (binary weight function), A064547; also A066724, A026477, A050376, A084400, A268386, A293439.
Note that A000069 and A001969, also A000201 and A001950 give other decompositions of the integers into two classes.
Cf. A124010 (prime exponents).
Cf. A026416, A026424, A059897.
Cf. A030230, A037445, A092248, A367514.

a000028 n = a000028_list !! (n-1)
a000028_list = filter (odd . sum . map a000120 . a124010_row) [1..]
-- Reinhard Zumkeller, Oct 05 2011

(Maple program from N. J. A. Sloane, Dec 20 2007) expts:=proc(n) local t1,t2,t3,t4,i; if n=1 then RETURN([0]); fi; if isprime(n) then RETURN([1]); fi; t1:=ifactor(n); if nops(factorset(n))=1 then RETURN([op(2,t1)]); fi; t2:=nops(t1); t3:=[]; for i from 1 to t2 do t4:=op(i,t1); if nops(t4) = 1 then t3:=[op(t3),1]; else t3:=[op(t3),op(2,t4)]; fi; od; RETURN(t3); end; # returns a list of the exponents e_1, e_2, ...
A000120 := proc(n) local w,m,i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end: # returns weight of binary expansion
LamMos:= proc(n) local t1,t2,t3,i; t1:=expts(n); add( A000120(t1[i]),i=1..nops(t1)); end; # returns sum of weights of exponents
M:=400; t0:=[]; t1:=[]; for n from 1 to M do if LamMos(n) mod 2 = 0 then t0:=[op(t0),n] else t1:=[op(t1),n]; fi; od: t0; t1; # t0 is A000379, t1 is the present sequence

iMoebiusMu[ n_ ] := Switch[ MoebiusMu[ n ], 1, 1, -1, -1, 0, If[ OddQ[ Plus@@ (DigitCount[ Last[ Transpose[ FactorInteger[ n ] ] ], 2, 1 ]) ], -1, 1 ] ]; q=Select[ Range[ 20000 ],iMoebiusMu[ # ]===-1& ] (* Wouter Meeussen, Dec 21 2007 *)
Rest[Select[Range[150],OddQ[Count[Flatten[IntegerDigits[#,2]&/@ Transpose[ FactorInteger[#]][[2]]],1]]&]] (* Harvey P. Dale, Feb 25 2012 *)

is(n)=my(f=factor(n)[,2]); sum(i=1,#f,hammingweight(f[i]))%2 \\ Charles R Greathouse IV, Aug 31 2013

Entry revised by N. J. A. Sloane, Dec 20 2007, restoring the original definition, correcting the entries and adding a new b-file.

A133009 One defining property of the sequences {A, B} = {A000069, A001969} is that they are the unique pair of sets complementary with respect to the nonnegative integers such that q(n) = |{x : x, y in A, x < y, x + y = n}| = |{x : x, y in B, x < y, x + y = n}| for all n >= 0. The present sequence gives the values of q(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 0, 1, 2, 1, 1, 2, 1, 1, 4, 1, 2, 3, 1, 3, 3, 2, 4, 3, 2, 3, 5, 2, 5, 5, 0, 5, 6, 3, 5, 5, 3, 4, 8, 4, 4, 6, 5, 5, 7, 6, 4, 7, 6, 5, 9, 5, 7, 8, 4, 7, 10, 7, 5, 10, 5, 5, 16, 5, 6, 11, 5, 9, 11, 8, 8, 10, 8, 8, 13, 7, 11, 12, 4, 12, 12, 8, 13, 10, 9, 11, 12, 10, 12, 12

Offset: 0

David W. Wilson, Dec 21 2007

nonn

David W. Wilson, Table of n, a(n) for n = 0..10000
Antoine Renard, Michel Rigo, and Markus A. Whiteland, q-Parikh Matrices and q-deformed binomial coefficients of words, arXiv:2402.05657 [cs.FL], 2024. See pp. 3, 12.
C. Sándor, Partitions of natural numbers and their representation functions, INTEGERS 4 (2004), #A18.
Jeffrey Shallit, Additive Number Theory via Automata and Logic, arXiv:2112.13627 [math.NT], 2021.

Cf. A000028, A000379, A000069, A001969, A133008.

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

A000028 Let k = p_1^e_1 p_2^e_2 p_3^e_3 ... be the prime factorization of n. Sequence gives k such that the sum of the numbers of 1's in the binary expansions of e_1, e_2, e_3, ... is odd.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Extensions

Views

Author

Keywords

Links

Crossrefs