A123193 -id:A123193 - cp's OEIS frontend

A130451 Number of divisors of A123193(n).

1, 2, 2, 3, 2, 2, 3, 2, 2, 5, 2, 2, 2, 8, 3, 2, 8, 2, 2, 8, 2, 8, 2, 2, 3, 2, 8, 8, 2, 2, 8, 2, 8, 2, 2, 8, 2, 5, 2, 8, 2, 2, 2, 8, 2, 8, 8, 2, 2, 8, 2, 8, 3, 2, 8, 8, 2, 8, 8, 2, 8, 2, 2, 2, 8, 8, 2, 2, 8, 2, 3, 8, 2, 8, 2, 2, 8, 8, 8, 8

Offset: 1

Giovanni Teofilatto, Aug 08 2007

isFib := proc(n) local i ; for i from 1 do if combinat[fibonacci](i) > n then RETURN(false) ; elif combinat[fibonacci](i) = n then RETURN(true) ; fi ; od: end: A123193 := proc(n) option remember ; local nmin,k ; nmin := 1 : if n > 1 then nmin := A123193(n-1)+1 ; fi ; for k from nmin do if isFib( numtheory[tau](k) ) then RETURN(k) ; fi ; od: end: A130451 := proc(n) numtheory[tau](A123193(n)) ; end: seq(A130451(n),n=1..80) ; # R. J. Mathar, Nov 16 2007

FibQ[n_] := IntegerQ[Sqrt[5 n^2 + 4]] || IntegerQ[Sqrt[5 n^2 - 4]];
Select[DivisorSigma[0, Range[250]], FibQ] (* Jean-François Alcover, Jan 27 2024 *)

a(n)=A000005(A123193(n)). - R. J. Mathar, Nov 16 2007

Corrected and extended by R. J. Mathar, Nov 16 2007

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.

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

A123240 Natural numbers with number of divisors not equal to a Fibonacci number.

Original entry on oeis.org

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, 60, 62, 63, 64, 65, 68, 69, 72, 74, 75, 76, 77, 80, 82, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 106, 108, 111, 112, 115, 116, 117, 118, 119, 120, 122, 123, 124, 125, 126, 129, 132, 133, 134, 140, 141, 142, 143, 144, 145, 146, 147, 148, 150, 153, 155, 156, 158, 159, 160, 161, 162, 164, 166, 168, 171, 172, 175, 176, 177, 178, 180, 183, 185, 187, 188, 192, 194, 196, 198, 200, 201, 202, 203, 204

Offset: 1

Giovanni Teofilatto, Oct 07 2006

No prime or square of a prime or square of a prime's square is included in this sequence. [From Avik Roy (avik_3.1416(AT)yahoo.co.in), Jan 23 2009]

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

Complement of A123193. Different from A000379 (e.g. contains 180).

If[MemberQ[Fibonacci[Range[20]],DivisorSigma[0,#]],Nothing,#]&/@Range[250] (* Harvey P. Dale, Mar 18 2023 *)

A348658 Numbers whose numerator and denominator of the harmonic mean of their divisors are both Fibonacci numbers.

Original entry on oeis.org

1, 3, 5, 6, 15, 21, 28, 140, 182, 496, 546, 672, 918, 1890, 2016, 4005, 4590, 24384, 52780, 55860, 68200, 84812, 90090, 105664, 145782, 186992, 204600, 381654, 728910, 907680, 1655400, 2302344, 2862405, 3828009, 3926832, 5959440, 21059220, 33550336, 33839988, 42325920

Offset: 1

Amiram Eldar, Oct 28 2021

nonn

Terms that also Fibonacci numbers are 1, 3, 5, 21, and no more below Fibonacci(300).

			3 is a term since the harmonic mean of its divisors is 3/2 = Fibonacci(4)/Fibonacci(3).
15 is a term since the harmonic mean of its divisors is 5/2 = Fibonacci(5)/Fibonacci(3).

Cf. A000045, A099377, A099378.

Similar sequences: A074266, A123193, A272412, A272440, A348659.

fibQ[n_] := Or @@ IntegerQ /@ Sqrt[{5 n^2 - 4, 5 n^2 + 4}]; h[n_] := DivisorSigma[0, n]/DivisorSigma[-1, n]; q[n_] := fibQ[Numerator[(hn = h[n])]] && fibQ[Denominator[hn]]; Select[Range[1000], q]

from itertools import islice
from sympy import integer_nthroot, gcd, divisor_sigma
def A348658(): # generator of terms
    k = 1
    while True:
        a, b = divisor_sigma(k), divisor_sigma(k,0)*k
        c = gcd(a,b)
        n1, n2 = 5*(a//c)**2-4, 5*(b//c)**2-4
        if (integer_nthroot(n1,2)[1] or integer_nthroot(n1+8,2)[1]) and (integer_nthroot(n2,2)[1] or integer_nthroot(n2+8,2)[1]):
            yield k
        k += 1
A348658_list = list(islice(A348658(),10)) # Chai Wah Wu, Oct 28 2021

A119885 Natural numbers with number of divisors equal to a Lucas number.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103

Offset: 1

Giovanni Teofilatto, Aug 01 2006

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

For natural numbers with number of divisors equal to a Fibonacci number, see A123193. [corrected by Matthew Vandermast, Oct 29 2008]

Cf. A000032.

lim = 250; t = LucasL /@ Range[0, lim]; Select[Range@ lim, MemberQ[t, DivisorSigma[0, #]] &]  (* Amiram Eldar, Sep 05 2019 after Michael De Vlieger at A123193 *)

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

cp's OEIS Frontend

A130451 Number of divisors of A123193(n).

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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.

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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.

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A123240 Natural numbers with number of divisors not equal to a Fibonacci number.

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A348658 Numbers whose numerator and denominator of the harmonic mean of their divisors are both Fibonacci numbers.

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A119885 Natural numbers with number of divisors equal to a Lucas number.

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

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A139588 Nonprime numbers with Fibonacci number of divisors.

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