A036537 -id:A036537 - cp's OEIS frontend

A056911 Odd squarefree numbers.

1, 3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 29, 31, 33, 35, 37, 39, 41, 43, 47, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 119, 123, 127, 129, 131, 133, 137, 139, 141, 143, 145, 149, 151

Offset: 1

James Sellers, Jul 07 2000

From Daniel Forgues, May 27 2009: (Start)
For any prime p, there are as many squarefree numbers having p as a factor as squarefree numbers not having p as a factor amongst all the squarefree numbers (one-to-one correspondence, both cardinality aleph_0).
E.g. there are as many even squarefree numbers as there are odd squarefree numbers.
For any prime p, the density of squarefree numbers having p as a factor is 1/p of the density of squarefree numbers not having p as a factor.
E.g. the density of even squarefree numbers is 1/p = 1/2 of the density of odd squarefree numbers (which means that 1/(p + 1) = 1/3 of the squarefree numbers are even and p/(p + 1) = 2/3 are odd). As a consequence the n-th even squarefree number is very nearly p = 2 times the n-th odd squarefree number (which means that the n-th even squarefree number is very nearly (p + 1) = 3 times the n-th squarefree number while the n-th odd squarefree number is very nearly (p + 1)/p = 3/2 the n-th squarefree number).
For any prime p, the n-th squarefree number not divisible by p is: n * (1 + 1/p) * zeta(2) + O(n^(1/2)) = n * (1 + 1/p) * (Pi^2 / 6) + O(n^(1/2)) (End)

			The exponents in the prime factorization of 15 are all equal to 1, so 15 appears here. The number 75 does not appear in this sequence, as it is divisible by the square number 25.

Zak Seidov, Table of n, a(n) for n = 1..12000
G. J. O. Jameson, Even and odd square-free numbers, Math. Gazette 94 (2010), 123-127; Author's copy.

Subsequence of A005117 and A036537.
Equals A039956/2.
Cf. A238711 (subsequence).

a056911 n = a056911_list !! (n-1)
a056911_list = filter ((== 1) . a008966) [1,3..]
-- Reinhard Zumkeller, Aug 27 2011

[n: n in [1..151 by 2] | IsSquarefree(n)]; // Bruno Berselli, Mar 03 2011

Select[Range[1,151,2],SquareFreeQ] (* Ant King, Mar 17 2013 *)

is(n)=n%2 && issquarefree(n) \\ Charles R Greathouse IV, Mar 26 2013

list(lim)=my(v=List()); forsquarefree(k=1,lim\1, if(k[1]%2, listput(v,k[1]))); Vec(v) \\ Charles R Greathouse IV, Jan 14 2025

A123314(A100112(a(n))) > 0. - Reinhard Zumkeller, Sep 25 2006
a(n) = n * (3/2) * zeta(2) + O(n^(1/2)) = n * (Pi^2 / 4) + O(n^(1/2)). - Daniel Forgues, May 27 2009
A008474(a(n)) * A000035(a(n)) = 1. - Reinhard Zumkeller, Aug 27 2011
Sum_{n>=1} 1/a(n)^s = ((2^s)* zeta(s))/((1+2^s)*zeta(2*s)). - Enrique Pérez Herrero, Sep 15 2012 [corrected by Amiram Eldar, Sep 26 2023]

A209061 Exponentially squarefree numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69

Offset: 1

Reinhard Zumkeller, Mar 13 2012

nonn

Numbers having only squarefree exponents in their canonical prime factorization.
According to the formula of Theorem 3 [Toth], the density of the exponentially squarefree numbers is 0.9559230158619... (A262276). - Peter J. C. Moses and Vladimir Shevelev, Sep 10 2015
From Vladimir Shevelev, Sep 24 2015: (Start)
A generalization. Let S be a finite or infinite increasing integer sequence s=s(n), s(0)=0.
Let us call a positive number N an exponentially S-number, if all exponents in its prime power factorization are in the sequence S.
Let {u(n)} be the characteristic function of S. Then, for the density h=h(S) of the exponentially S-numbers, we have the representations
h(S) = Product_{prime p} Sum_{j in S} (p-1)/p^(j+1) = Product_{p} (1 + Sum_{j>=1} (u(j) - u(j-1))/p^j). In particular, if S = {0,1}, then the exponentially S-numbers are squarefree numbers; if S consists of 0 and {2^k}A138302%20(see%20%5BShevelev%5D,%202007);%20if%20S%20consists%20of%200%20and%20squarefree%20numbers,%20then%20u(n)=%7Cmu(n)%7C,%20where%20mu(n)%20is%20the%20M%C3%B6bius%20function%20(A008683),%20we%20obtain%20the%20density%20h%20of%20the%20exponentially%20squarefree%20numbers%20(cf.%20Toth's%20link,%20Theorem%203);%20the%20calculation%20of%20h%20with%20a%20very%20high%20degree%20of%20accuracy%20belongs%20to%20_Juan%20Arias-de-Reyna">{k>=0}, then the exponentially S-numbers form A138302 (see [Shevelev], 2007); if S consists of 0 and squarefree numbers, then u(n)=|mu(n)|, where mu(n) is the Möbius function (A008683), we obtain the density h of the exponentially squarefree numbers (cf. Toth's link, Theorem 3); the calculation of h with a very high degree of accuracy belongs to _Juan Arias-de-Reyna (A262276). Note that if S contains 1, then h(S) >= 1/zeta(2) = 6/Pi^2; otherwise h(S) = 0. Indeed, in the latter case, the density of the sequence of exponentially S-numbers does not exceed the density of A001694, which equals 0. (End)
The term "exponentially squarefree number" was apparently coined by Subbarao (1972). - Amiram Eldar, May 28 2025

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Xiaodong Cao and Wenguang Zhai, Some arithmetic functions involving exponential divisors, JIS 13 (2010), Article 10.3.7.
Y.-F. S. Petermann, Arithmetical functions involving exponential divisors: note on two papers by L. Toth, Ann. Univ. Sci. Budapest, Sect. Comp. 32 (2010) 143-149.
Vladimir Shevelev, Compact integers and factorials, Acta Arithmetica 126:3 (2007), pp. 195-236.
Vladimir Shevelev, Exponentially S-numbers, arXiv:1510.05914 [math.NT], 2015.
Vladimir Shevelev, Set of all densities of exponentially S-numbers, arXiv:1511.03860 [math.NT], 2015.
Vladimir Shevelev, A fast computation of density of exponentially S-numbers, arXiv:1602.04244 [math.NT], 2016.
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175(2016), 385-395.
H. M. Stark, On the asymptotic density of the k-free integers, Proc. Amer. Soc. 17 (1966), 1211-1214.
M. V. Subbarao, On some arithmetic convolutions, in: A. A. Gioia and D. L. Goldsmith (eds.), The Theory of Arithmetic Functions: Proceedings of the Conference at Western Michigan University, April 29 - May 1, 1971, Berlin, Heidelberg: Springer Berlin Heidelberg, 2006, pp. 247-271; alternative link.
László Tóth, On certain arithmetic functions involving exponential divisors, II., Annales Univ. Sci. Budapest., Sect. Comp., 27 (2007), 155-166 and arXiv:0708.3557 [math.NT], 2007-2009.

Complement of A130897.
A005117, A004709, and A046100 are subsequences.
Cf. A001221, A001694, A008683, A008966, A036537, A115063, A138302, A197680, A262276, A262675, A268335, A270428.

a209061 n = a209061_list !! (n-1)
a209061_list = filter
   (all (== 1) . map (a008966 . fromIntegral) . a124010_row) [1..]

Select[Range@ 69, Times @@ Boole@ Map[SquareFreeQ, Last /@ FactorInteger@ #] > 0 &] (* Michael De Vlieger, Sep 07 2015 *)

is(n)=my(f=factor(n)[,2]); for(i=1,#f,if(!issquarefree(f[i]), return(0))); 1 \\ Charles R Greathouse IV, Sep 02 2015

A166234(a(n)) <> 0.
Product_{k=1..A001221(n)} A008966(A124010(n,k)) = 1.
One can prove that the principal term of Toth's asymptotics for the density of this sequence (cf. Toth's link, Theorem 3) equals also Product_{prime p}(Sum_{j in S}(p-1)/p^{j+1})*x, where S is the set of 0 and squarefree numbers. The remainder term O(x^(0.2+t)), where t>0 is arbitrarily small, was obtained by L. Toth while assuming the Riemann Hypothesis. - Vladimir Shevelev, Sep 12 2015

A048109 Numbers having equally many squarefree and nonsquarefree divisors; number of unitary divisors of n (A034444) = number of non-unitary divisors of n (A048105).

Original entry on oeis.org

8, 24, 27, 40, 54, 56, 88, 104, 120, 125, 135, 136, 152, 168, 184, 189, 232, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 375, 376, 378, 408, 424, 440, 456, 459, 472, 488, 513, 520, 536, 552, 568, 584, 594, 616, 621, 632, 664, 680, 686, 696

Offset: 1

Labos Elemer

nonn

For these terms the number of divisors should be a special power of two because ud(n) = 2^r and nud(n) = ud(n). In particular the exponent of 2 is 1+A001221(n), the number of distinct prime factors + 1. Thus this is a subsequence of A036537 where A000005(A036537(n)) = 2^s; here s = 1+A001221(n).
Let us introduce a function D(n) = sigma_0(n)/2^(alpha(1)+...+alpha(r)), sigma_0(n) number of divisors of n (A000005), prime factorization of n = p(1)^alpha(1) * ... * p(r)^alpha(r), alpha(1)+...+alpha(r) is sequence (A001222). This function splits the set of positive integers into subsets, according to the value of D(n). Squarefree numbers (A005117) has D(n)=1, other numbers are "deviated" from the squarefree ideal and have 0 < D(n) < 1. So for D(n)=1/2 we have A048109, D(n)=3/4 we have A060687. - Ctibor O. Zizka, Sep 21 2008
Integers n such that there are exactly 3 Abelian groups of order n. That is, n such that A000688(n)=3. In other words, in the prime factorization of n there is exactly one prime with exponent of 3 and the others have exponent of 1. - Geoffrey Critzer, Jun 09 2015
The asymptotic density of this sequence is (6/Pi^2) * Sum_{k>=1} 1/(prime(k)^2*(prime(k)+1)) = (1/zeta(2)) * Sum_{k>=3} (-1)^(k+1) * P(k) = 0.0741777413672596019212880156082745910562809066233004356300970463709875..., where P is the prime zeta function. - Amiram Eldar, Jul 11 2020

			n = 88 = 2*2*2*11 has 8 divisors, of which 4 are unitary divisors (because of 2 distinct prime factors) and 4 are nonunitary divisors: U={1,88,11,8} and NU = {2,44,4,22}.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A001221, A034444, A036537, A048106, A048107.

filter:= proc(n) local F;
  F:= ifactors(n)[2];
  mul(t[2]+1,t=F) = 2^(1+nops(F))
end proc;
select(filter, [$1..1000]); # Robert Israel, Jun 09 2015

Position[Table[FiniteAbelianGroupCount[n], {n, 1, 1000}],3] // Flatten (* Geoffrey Critzer, Jun 09 2015 *)

is(n)=select(e->e>1, factor(n)[,2])==[3]~ \\ Charles R Greathouse IV, Jun 10 2015

isok(n) = sumdiv(n, d, issquarefree(d)) == sumdiv(n, d, !issquarefree(d)); \\ Michel Marcus, Jun 24 2015

from math import isqrt
from sympy import mobius, primerange
def A048109(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x): return sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def f(x): return int(n+x+sum(sum(-g(x//p**j) if j&1 else g(x//p**j) for j in range(3,x.bit_length())) for p in primerange(isqrt(x)+1)))
    return bisection(f,n,n) # Chai Wah Wu, Feb 24 2025

Numbers k such that d(k) = 2^(omega(k)+1) or A000005(k) = 2^(A001221(k) + 1) = 2 * A034444(k).

New name based on comment by Ivan Neretin, Jun 19 2015

A337050 Numbers without an exponent 2 in their prime factorization.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 48, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87

Offset: 1

Amiram Eldar, Aug 12 2020

Numbers k such that the powerful part (A057521) of k is a cubefull number (A036966).
Numbers k such that A003557(k) = k/A007947(k) is a powerful number (A001694).
The asymptotic density of this sequence is Product_{primes p} (1 - 1/p^2 + 1/p^3) = 0.748535... (A330596).
A304364 is apparently a subsequence.
These numbers were named semi-2-free integers by Suryanarayana (1971). - Amiram Eldar, Dec 29 2020

			6 = 2^1 * 3^1 is a term since none of the exponents in its prime factorization is equal to 2.
9 = 3^2 is not a term since it has an exponent 2 in its prime factorization.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ertan Elma and Greg Martin, Distribution of the number of prime factors with a given multiplicity, Canadian Mathematical Bulletin, Vol. 67, No. 4 (2024), pp. 1107-1122; arXiv preprint, arXiv:2406.04574 [math.NT], 2024.
D. Suryanarayana, Semi-k-free integers, Elemente der Mathematik, Vol. 26 (1971), pp. 39-40.
D. Suryanarayana and R. Sitaramachandra Rao, Distribution of semi-k-free integers, Proceedings of the American Mathematical Society, Vol. 37, No. 2 (1973), pp. 340-346.

Complement of A038109.
Cf. A003557, A007947, A057521, A304364, A330596.
A005117, A036537, A036966, A048109, A175496, A268335 and A336590 are subsequences.
Numbers without an exponent k in their prime factorization: A001694 (k=1), this sequence (k=2), A386799 (k=3), A386803 (k=4), A386807 (k=5).
Numbers that have exactly m exponents in their prime factorization that are equal to 2: this sequence (m=0), A386796 (m=1), A386797 (m=2), A386798 (m=3).

q:= n-> andmap(i-> i[2]<>2, ifactors(n)[2]):
select(q, [$1..100])[];  # Alois P. Heinz, Aug 12 2020

Select[Range[100], !MemberQ[FactorInteger[#][[;;, 2]], 2] &]

is(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 2] == 2, return(0))); 1; } \\ Amiram Eldar, Oct 21 2023

Sum_{n>=1} 1/a(n)^s = zeta(s) * Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s)), for s > 1. - Amiram Eldar, Oct 21 2023

A046642 Numbers k such that k and number of divisors d(k) are relatively prime.

Original entry on oeis.org

1, 3, 4, 5, 7, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 64, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 100, 101, 103, 105, 107, 109, 111, 113, 115, 119, 121, 123, 125, 127, 129, 131

Offset: 1

Labos Elemer

Numbers k such that tau(k)^phi(k) == 1 (mod k), where tau(k) is the number of divisors of k (A000005) and phi(k) is the Euler phi function (A000010). - Michel Lagneau, Nov 20 2012
Density is at least 4/Pi^2 = 0.405... since A056911 is a subsequence, and at most 1/2 since all even numbers in this sequence are squares. The true value seems to be around 0.4504. - Charles R Greathouse IV, Mar 27 2013
They are called anti-tau numbers by Zelinsky (see link) and their density is at least 3/Pi^2 (theorem 57 page 15). - Michel Marcus, May 31 2015
From Amiram Eldar, Feb 21 2021: (Start)
Spiro (1981) proved that the number of terms of this sequence that do not exceed x is c * x + O(sqrt(x)*log(x)^3), where 0 < c < 1 is the asymptotic density of this sequence.
The odd numbers whose number of divisors is a power of 2 (the odd terms of A036537) are terms of this sequence. Their asymptotic density is A327839/A076214 = 0.4212451116... which is a better lower bound than 4/Pi^2 for the asymptotic density of this sequence.
A better upper limit than 0.5 can be obtained by considering the subsequence of odd numbers whose 3-adic valuation is not of the form 3*k-1 (i.e., odd numbers without those k with gcd(k, tau(k)) = 3), whose asymptotic density is 6/13 = 0.46153...
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 5, 49, 459, 4535, 45145, 450710, 4504999, 45043234, 450411577, 4504050401, ... (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Mart Abel, Helena Lauer, and Ellen Redi, About the number of τ-numbers relative to polynomials with integer coefficients, Acta Comment. Univ. Tartu. Math. 25, No. 1, 107-117, 2021.
Claudia A. Spiro, The Frequency with Which an Integral-Valued, Prime-Independent, Multiplicative or Additive Function of n Divides a Polynomial Function of n, Ph. D. Thesis, University of Illinois, Urbana-Champaign, 1981.
Joshua Zelinsky, Tau Numbers: A Partial Proof of a Conjecture and Other Results, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.

Cf. A000005, A000010, A009191, A009230, A036537, A056911, A076214, A327839.

a046642 n = a046642_list !! (n-1)
a046642_list = map (+ 1) $ elemIndices 1 a009191_list
-- Reinhard Zumkeller, Aug 14 2011

Select[ Range[200], CoprimeQ[#, DivisorSigma[0, #]] &] (* Jean-François Alcover, Oct 20 2011 *)

is(n)=gcd(numdiv(n),n)==1 \\ Charles R Greathouse IV, Mar 27 2013

A009191(a(n)) = 1.

A162643 Numbers whose number of divisors is not a power of 2.

Original entry on oeis.org

4, 9, 12, 16, 18, 20, 25, 28, 32, 36, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 90, 92, 96, 98, 99, 100, 108, 112, 116, 117, 121, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 160, 162, 164, 169, 171, 172, 175, 176, 180, 188, 192, 196, 198

Offset: 1

Reinhard Zumkeller, Jul 08 2009

nonn

A number m is a term if and only if it has at least one non-infinitary divisor, or A000005(m) > A037445(m). - Vladimir Shevelev, Feb 23 2017

The asymptotic density of this sequence is 1 - A327839 = 0.3121728605... - Amiram Eldar, Jul 28 2020

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Complement of A036537.
A072587 is a subsequence.
Cf. A000005, A037445, A209229, A327839

a162643 n = a162643_list !! (n-1)
a162643_list = filter ((== 0) . a209229 . a000005) [1..]
-- Reinhard Zumkeller, Nov 15 2012

Select[Range@ 192, ! IntegerQ@ Log2@ DivisorSigma[0, #] &] (* Michael De Vlieger, Feb 24 2017 *)

from itertools import count, islice
from sympy import factorint
def A162643_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:any(map(lambda m:((k:=m+1)&-k)^k,factorint(n).values())),count(max(startvalue,1)))
A162643_list = list(islice(A162643_gen(),30)) # Chai Wah Wu, Jan 04 2023

A209229(A000005(a(n))) = 0. - Reinhard Zumkeller, Nov 15 2012

A002035 Numbers that contain primes to odd powers only.

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95, 96, 97, 101

Offset: 1

N. J. A. Sloane

Complement of the union of {1} and A072587. - Reinhard Zumkeller, Nov 15 2012, corrected version from Jun 23 2002
A036537 is a subsequence and this sequence is a subsequence of A162644. - Reinhard Zumkeller, Jul 08 2009
The asymptotic density of this sequence is A065463. - Amiram Eldar, Sep 18 2022

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).

T. D. Noe, Table of n, a(n) for n = 1..1000
Eckford Cohen, Quadratic congruences with an odd number of summands, Amer. Math. Monthly, 73 (1966), 138-143.

Cf. A000203, A036537, A065463, A072586, A072587, A124010, A162644, A188999, A295316.

a002035 n = a002035_list !! (n-1)
a002035_list = filter (all odd . a124010_row) [1..]
-- Reinhard Zumkeller, Nov 14 2012

isA002035 := proc(n)
    local pe;
    for pe in ifactors(n)[2] do
        if type(pe[2],'even') then
            return false;
        end if;
    end do:
    true ;
end proc:
A002035 := proc(n)
    option remember;
    if n =1 then
        2;
    else
        for a from procname(n-1)+1 do
            if isA002035(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A002035(n),n=1..100) ; # R. J. Mathar, Nov 27 2017

ok[n_] := And @@ OddQ /@ FactorInteger[n][[All, 2]];
Select[Range[2, 101], ok]
(* Jean-François Alcover, Apr 22 2011 *)
Select[Range[2,110],AllTrue[FactorInteger[#][[All,2]],OddQ]&] (* Harvey P. Dale, Nov 02 2022 *)

is(n)=Set(factor(n)[,2]%2)==[1] \\ Charles R Greathouse IV, Feb 07 2017

More terms from Reinhard Zumkeller, Jun 23 2002

A348271 a(n) is the sum of noninfinitary divisors of n.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 8, 0, 0, 0, 14, 0, 9, 0, 12, 0, 0, 0, 0, 5, 0, 0, 16, 0, 0, 0, 12, 0, 0, 0, 41, 0, 0, 0, 0, 0, 0, 0, 24, 18, 0, 0, 56, 7, 15, 0, 28, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 24, 42, 0, 0, 0, 36, 0, 0, 0, 45, 0, 0, 20, 40, 0, 0, 0, 84, 39

Offset: 1

Amiram Eldar, Oct 09 2021

nonn

			a(12) = 8 since 12 has 2 noninfinitary divisors, 2 and 6, and 2 + 6 = 8.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Infinitary Divisor.

Cf. A000203, A036537, A049417, A077609, A162643, A327633.

Similar sequences: A034448, A048146, A051377, A188999.

f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; a[n_]:= DivisorSigma[1,n] - isigma[n]; Array[a, 100]

a(n) = A000203(n) - A049417(n).
a(n) = 0 if and only if the number of divisors of n is a power of 2, (i.e., n is in A036537).
a(n) > 0 if and only if the number of divisors of n is not a power of 2, (i.e., n is in A162643).

A162511 Multiplicative function with a(p^e) = (-1)^(e-1).

Original entry on oeis.org

1, 1, 1, -1, 1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -1, 1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, 1, 1, -1, -1, -1, 1, -1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, -1, -1, 1, 1, 1, -1, 1, 1, 1, -1, 1, 1, -1, -1, 1, 1, 1, -1, -1, 1, 1, -1, 1, 1, 1, 1, 1, -1, 1, -1

Offset: 1

Gerard P. Michon, Jul 05 2009

G. C. Greubel, Table of n, a(n) for n = 1..5000
Gérard P. Michon, Multiplicative functions.

Cf. A005117, A076479, A162510, A162512, A002035, A072587, A036537, A162643, A162644, A162645, A046660, A008836, A307868.

A162511 := proc(n)
    local a,f;
    a := 1;
    for f in ifactors(n)[2] do
        a := a*(-1)^(op(2,f)-1) ;
    end do:
    return a;
end proc: # R. J. Mathar, May 20 2017

a[n_] := (-1)^(PrimeOmega[n] - PrimeNu[n]); Array[a, 100] (* Jean-François Alcover, Apr 24 2017, after Reinhard Zumkeller *)

a(n)=my(f=factor(n)[,2]); prod(i=1,#f,-(-1)^f[i]) \\ Charles R Greathouse IV, Mar 09 2015

from sympy import factorint
from operator import mul
def a(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [(-1)**(f[i] - 1) for i in f]) # Indranil Ghosh, May 20 2017

from functools import reduce
from sympy import factorint
def A162511(n): return -1 if reduce(lambda a,b:~(a^b), factorint(n).values(),0)&1 else 1 # Chai Wah Wu, Jan 01 2023

Multiplicative function with a(p^e)=(-1)^(e-1) for any prime p and any positive exponent e.
a(n) = 1 when n is a squarefree number (A005117).
From Reinhard Zumkeller, Jul 08 2009 (Start)
a(n) = (-1)^(A001222(n)-A001221(n)).
a(A162644(n)) = +1; a(A162645(n)) = -1. (End)
a(n) = A076479(n) * A008836(n). - R. J. Mathar, Mar 30 2011
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A307868. - Amiram Eldar, Sep 18 2022
Dirichlet g.f.: Product_{p prime} ((p^s + 2)/(p^s + 1)). - Amiram Eldar, Oct 26 2023

A348341 a(n) is the number of noninfinitary divisors of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 2, 0, 2, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 6, 1, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 2, 3, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 2, 2, 0, 0, 0, 6, 3, 0, 0, 4, 0, 0, 0

Offset: 1

Amiram Eldar, Oct 13 2021

nonn

			a(4) = 1 since 4 has one noninfinitary divisor, 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Infinitary Divisor.

Cf. A000005, A036537, A037445, A077609, A162643, A327576, A348271.

Similar sequences: A034444, A048105, A286324, A049419.

a[1] = 0; a[n_] := DivisorSigma[0, n] - Times @@ Flatten[2^DigitCount[#, 2, 1] & /@ FactorInteger[n][[;; , 2]]]; Array[a, 100]

A348341(n) = (numdiv(n)-factorback(apply(a -> 2^hammingweight(a), factorint(n)[, 2]))); \\ Antti Karttunen, Oct 13 2021

a(n) = A000005(n) - A037445(n).
a(n) = 0 if and only if the number of divisors of n is a power of 2, (i.e., n is in A036537).
a(n) > 0 if and only if the number of divisors of n is not a power of 2, (i.e., n is in A162643).
Sum_{k=1..n} a(k) ~ c * n * log(n), where c = (1 - 2 * A327576) = 0.266749... . - Amiram Eldar, Dec 09 2022

cp's OEIS Frontend

A056911 Odd squarefree numbers.

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A048109 Numbers having equally many squarefree and nonsquarefree divisors; number of unitary divisors of n (A034444) = number of non-unitary divisors of n (A048105).

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A337050 Numbers without an exponent 2 in their prime factorization.

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A046642 Numbers k such that k and number of divisors d(k) are relatively prime.

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A162643 Numbers whose number of divisors is not a power of 2.

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