A123240 -id:A123240 - cp's OEIS frontend

A130331 Number of divisors of A123240(n).

4, 4, 4, 6, 4, 4, 6, 6, 4, 4, 4, 4, 6, 6, 4, 4, 4, 9, 4, 4, 6, 6, 4, 10, 6, 4, 6, 4, 4, 4, 12, 4, 6, 7, 4, 6, 4, 12, 4, 6, 6, 4, 10, 4, 12, 4, 4, 4, 12, 4, 6, 4, 4, 4, 12, 6, 6, 9, 4, 12, 4, 10, 4, 6, 6, 4, 4, 16, 4, 4, 6, 4, 12, 4, 12, 4, 4, 12, 4, 4, 4, 15, 4, 4, 6, 6, 12, 6, 4, 12, 4, 4, 12, 4, 10, 6, 4

Offset: 1

Giovanni Teofilatto, Aug 07 2007

By definition, each term is in A001690. - R. J. Mathar, Jan 15 2008

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A000379, A123240.

seq[max_] := Module[{d = DivisorSigma[0, Range[max]], dmax, k = 2}, dmax = Max[d]; While[Fibonacci[k] < dmax, k++]; Select[d, !MemberQ[Fibonacci[Range[2, k]], #] &]]; seq[300] (* Amiram Eldar, Apr 08 2024 *)

a(n) = A000005(A123240(n)). - R. J. Mathar, Jan 15 2008

More terms from R. J. Mathar, Jan 15 2008

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.

Original entry on oeis.org

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.

A123193 Natural numbers with number of divisors equal to a Fibonacci number.

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, 149, 151, 152, 154, 157, 163, 165, 167, 169, 170, 173, 174, 179, 181, 182, 184, 186, 189, 190, 191, 193, 195, 197, 199, 211, 222, 223, 227, 229, 230, 231, 232, 233, 238, 239, 241, 246, 248, 250

Offset: 1

Giovanni Teofilatto, Oct 04 2006

nonn

Does not contain (for example) 180, so is different from A000028. - Max Alekseyev, Sep 20 2007
How dense is this sequence? There are 7 members up to 10^1, 42 up to 10^2, 364 up to 10^3, 3379 up to 10^4, 31864 up to 10^5, 303623 up to 10^6, 2907125 up to 10^7, 27893864 up to 10^8, and 268099330 up to 10^9. - Charles R Greathouse IV, Sep 16 2015
Partial answer: a(n) << n log n/(log log n)^k for any k. Proof: Since 0 is a Fibonacci number, and Fibonacci numbers are periodic mod any number, 2^(k+1) divides infinitely many Fibonacci numbers. Take some positive Fibonacci number F divisible by 2^(k+1). By Landau's theorem there are >> x (log log x)^k/log x odd squarefree numbers divisible by k+1 primes up to x. Multiply each by 2^(F/2^(k+1)-1) which leaves the density unchanged since the expression is constant in k, and note that the products have exactly F divisors. - Charles R Greathouse IV, Sep 16 2015

T. D. Noe, Table of n, a(n) for n = 1..10000

Complement of A123240. Different from A026416.

lim = 250; t = Fibonacci /@ Range@ lim; Select[Range@ lim, MemberQ[t, DivisorSigma[0, #]] &] (* Michael De Vlieger, Sep 16 2015 *)

is(n)=my(k=numdiv(n)^2); issquare(k+=(k+1)<<2)||issquare(k-8) \\ Charles R Greathouse IV, Sep 16 2015

A122895 Characteristic function of natural numbers with number of divisors equal to a Fibonacci number.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1

Offset: 1

Giovanni Teofilatto, Oct 24 2006

Cf. A000005, A010056, A115568, A123193, A123240.

fibQ[n_] := IntegerQ@ Sqrt[5*n^2+4] || IntegerQ@ Sqrt[5*n^2-4]; Boole[ fibQ /@ DivisorSigma[0, Range[103]]] (* Giovanni Resta, Mar 10 2017 *)

isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8));
a(n) = isfib(numdiv(n)); \\ Michel Marcus, Mar 10 2017

from sympy import divisor_count
from sympy.ntheory.primetest import is_square
def A122895(n): return int(is_square(m:=5*int(divisor_count(n))**2-4) or is_square(m+8)) # Chai Wah Wu, Oct 10 2023

a(n) = A010056(A000005(n)). - Chayim Lowen, Aug 01 2015

a(0)=0 removed from data by Michel Marcus, Mar 10 2017

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A130331 Number of divisors of A123240(n).

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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.

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A123193 Natural numbers with number of divisors equal to a Fibonacci number.

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A122895 Characteristic function of natural numbers with number of divisors equal to a Fibonacci number.

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Mathematica

PARI

Python

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Extensions